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THE 



CONCEPTION OF THE INFINITE, 



AND THE 



SOLUTION OF THE MATHEMATICAL 
ANTINOMIES: 



A STUDY IN PSYUHOLDiilCAL ANALYSIS. 



GEORGE S. FULLERTON, A.M., B.D., 

ADJUNCT PROFESSOR OF PHILOSOPHY IN THE UNIVERSITY OF PENNSYLVANIA. 






PHILADELPHIA: 

J. B. LIPPINCOTT COMPANY. 

188 7. 






Copyright, 1887, by George S. Fiillertox. 



PEEFAOE. 



The question treated in this little volume is 
one of no small interest from several quite 
different points of view. To one interested in 
lucid and systematic thinking, the tangle of 
thought which has always obtained in this 
corner of the philosophic field cannot but be 
repulsive and irritating. To be told that of 
two impossible things one must be true; that 
of the same two lines one may be looked upon 
as, at pleasure, equal to, less than, or greater 
than the other, both remaining unchanged ; 
that Achilles, running rapidly, can never over- 
take the tortoise, moving slowly ; to be told 
all this seriously, by men whose calling it is 
to think and to teach others to think, is well 
calculated to bring not merely suspicion but 
contempt upon speculative thought, and de- 
servedly. Who has not puzzled, on his first 



4 PREFACE. 

introduction to Logic, over some of these an- 
tinomies, and been silenced unconvinced by the 
practical demonstration, — as by walking, in the 
case of the argument against motion, — which 
cuts the knot but does not solve it, leaving in 
the mind a disagreeable sense that the argu- 
ment must be wrong somewhere, and yet a 
consciousness that it certainly seems perfectly 
sound ? When the metaphysician proves to us 
that a rhinoceros is a mosquito, his chain of 
reasoning is rendered innocuous by the striking 
incongruity of the conclusion ; but if we ob- 
serve no flaw in his reasoning, we cannot help 
recognizing the perplexing truth that it is the 
experienced fact alone which has prevented as- 
sent, and that a precisely similar argument, the 
conclusion of which cannot be similarly tested, 
may yet induce assent, though equally errone- 
ous. If we have no better reason for rejecting 
an argument, what can be our criterion when 
we leave the sphere of the immediately palpa- 
ble ? He who has convinced himself that the 
minute hand of a clock cannot overtake the 
hour hand, will be enlightened when the clock 
strikes at noon ; but he who has followed Mr. 
Spencer into his discussions regarding our no- 



PREFACE. 5 

tions of infinite space or time, will be filled 
with inward dismay if he hang his hope upon 
any such practical expedient. Civil history can- 
not be studied in the laboratory, nor erroneous 
ideas as to infinite space rectified with the aid 
of the foot-rule. In this sphere, too, the ques- 
tion of a careful and thorough analysis of our 
conception of the infinite is of more than a 
merely intellectual interest, and any erroneous 
conception which can blossom out into such a 
development as the " Philosophy of the Condi- 
tioned," with its implications, has a religious 
significance which cannot be overlooked. The 
analysis of this single conception is, moreover, 
of importance as throwing light upon the pro- 
cedure of thought in general, and will to many 
be of more interest in this connection than for 
its own sake. I have endeavored to write with 
extreme clearness and simplicity, and to avoid, 
as much as possible, all issues not directly con- 
nected with the immediate subject; and whether 
my discussion meet with assent or dissent, I do 
not think it will be charged with the obscurity 
characteristic of discussions upon this much- 
mooted topic. 

Portions of the book are reprinted, with ad- 
l* 



6 PREFACE. 

ditions and alterations, from the American Jour- 
nal of Speculative Philosophy and from the Brit- 
ish periodical Mind, in which they originally 
appeared. 

University of Pennsylvania, December, 1886. 



CONTENTS. 



CHAPTEK I. 

PAGE 



Introductory 



CHAPTEK II. 
The Conception not Quantitative . . . .20 

CHAPTEK III. 
The Antinomies of Hamilton 34 

CHAPTER IV. 
Kant, Mill, and Clifford 53 

CHAPTEK V. 

The Conceivable and the Existent . . 77 

CHAPTE It V I. 
The Conceivability of the Infinite . . .90 



THE 

CONCEPTION OF THE INFINITE, 



SOLUTION OF THE MATHEMATICAL 
ANTINOMIES : 

A STUDY IN PSYCHOLOGICAL ANALYSIS. 



CHAPTER I. 

INTRODUCTORY. 

The doctrine that there may be an infinity of 
worlds, thought Plutarch,* is to be repudiated. 
Providence could not possibly take charge of so 
many; "troublesome and boundless infinity" 
could be grasped by no consciousness. 

Plutarch's decision as to the unknowability 
of the infinite the student of the history of 
thought will find reiterated by thinkers who 
agree upon little else than the one point, that 

* " De defectu oraculorum," c. 24. 



10 THE CONCEPTION OF THE INFINITE. 

when we leave the finite and talk of the infinite 
we are playing with a word, — deceiving our- 
selves into believing that we can know what is 
in its very nature inconceivable. 

The notion of the infinite, it is said,* is a 
negative one : all our experience of objects 
being of them as finite, we can think to our- 
selves the negation of this condition, and thus 
form a negative conception, — i.e., not an affir- 
mation of a quality or attribute, but a simple 
denial of a quality known. But to know 
an object — for example, space — as infinite, 
this is beyond our power. The human mind 
is finite; all. which can become an object of 
consciousness is finite; and "troublesome and 
boundless infinity" cannot be an object of 
thought. 

" Whereas, for this very reason," said Cud- 
worth, f " because more could be added to the 
magnitude of the corporeal world infinitely, or 
without end, therefore it is impossible, that it 
should ever be positively and actually infinite; 

* See Hamilton's Discussions, " Philosophy of the Uncon- 
ditioned." 

f "Intellectual System," chap, v., Andover, 1838, vol. ii. 
p. 45. 



INTRODUCTORY. \\ 

that is such as to which nothing more can pos- 
sibly be added." 

A favorite position : all attempts at cognizing 
the infinite must result in the indefinite; for 
the infinite can only be known by the progres- 
sive addition of finites, an addition which can 
be completed only in an infinite time ; in other 
words, can never be completed. Each stage in 
the addition gives but the indefinite, and the 
last stage, the infinite, is, by the terms of the 
problem, unattainable. 

How, indeed, it is asked, could a finite mind 
know the infinite? "Adequately to know what 
is infinite is to have infinite knowledge."* We 
cannot, surely, lay claim to that. 

The famous antinomies of the philosopher of 
Konigsberg have, more than anything else, 
stirred up the minds of men to a consideration 
of the problem. The indefinite, said Kant, we 
may know as a whole, not by passing -succes- 
sively over each of its parts, but immediately 
as a unit, by means of its limits ; but in the case 
of the infinite, since there are no limits, we 
must arrive at our cognition by a successive 

* " The Battle of the Two Philosophies," p. 24. 



12 THE CONCEPTION OF THE INFINITE. 

addition of parts, which addition must neces- 
sarily be itself endless, and therefore the attempt 
thus to know the infinite futile. Although we 
are unable to conceive of an absolute commence- 
ment to time or of an absolute limit to space, 
we are not on that account justified in assuming 
either to be infinite, as this we can only know 
when we have passed successively over all spaces 
and all times. The whole question is one the 
decision of which is beyond the scope of human 
reason; both alternatives are equally inconceiv- 
able. 

The doctrine of the inconceivability of the 
infinite is much dwelt upon by Sir William 
Hamilton as one phase of his cherished theory, 
the Philosophy of the Conditioned. He argues 
that space, for example, cannot be cognized as 
either infinitely extended and infinitely divisible, 
or, on the other hand, as absolutely limited, 
whether as a maximum or minimum, just as Kant 
argued. Mr. Mansel has accepted and de- 
fended * the positions taken by Hamilton ; while 
the same arguments, which are used by Sir 



* See Mr. Mansel's "Philosophy of the Conditioned." 
London: 1866. 



INTRODUCTORY. 13 

William Hamilton and his follower, Mr. Man- 
sel, to elevate faith at the expense of reason or 
science, are to be found upon the pages of 
the " First Principles of Philosophy," where 
they are used hy Mr. Spencer in support of a 
philosophy widely different from the Hamil- 
tonian. 

It would not he difficult to multiply testimo- 
nies to the .inconceivability of the infinite, for 
the misconception which we find in Plutarch 
appears and reappears in divers forms in dif- 
ferent ages and climes, much as the Wandering 
Jew jaarp be supposed to have presented him- 
self; and, to carry out the simile, the well-worn 
dress in which the doctrine usually comes to the 
surface — the statement that we can know the 
infinite only hy an endless addition of finites — 
may he not inaptly compared to the threadbare 
garment on his hack. 

It would seem surprising that there could he 
so universal a misconception with regard to the 
nature of a conception actually present at some 
time or other in the mind probably of every 
man, and in the case of many not unfrequently 
present; a conception sufficiently familiar and 
important to be recognized and marked by its 



14 THE CONCEPTION OF THE INFINITE. 

appropriate name. But a very little knowledge 
of psychological processes will make one cogni- 
zant of the fact that the interpretation of con- 
sciousness is by no means the easy and simple 
task that by a novice it might be supposed 
to be. 

There are but few who have an analytical 
knowledge of even the most common of their 
mental operations ; and the average man is very 
literally incapable of telling what may at any 
moment be passing in his mind. It is a matter 
of surprise, for instance, to one unaccustomed to 
psychological analysis, to learn that distance is 
not directly perceived by vision, but that judg- 
ments of distance are the result of a rapid pro- 
cess of reasoning, and imply a generalization 
from past experience. The judgment seems in- 
stantaneous and intuitive. Those elements which 
are in fact visual and actually present are not 
distinguished from those present only by sug- 
gestion ; that is, present in the imagination. 
The mental state is grasped as a unit, and for 
practical purposes no analysis into its elements 
is necessary. So it is with the concept, or gen- 
eral notion. That men form general notions, or 
at least represent in mind objects and their re- 



INTR OD UCTOR Y. 15 

lations in some way different from that in which 
simple intuitions are singly represented, all are 
ready to acknowledge. But that the psychical 
elements concerned in this act are not clearly 
apprehended is evident from the common war- 
fare of Realism, Nominalism, and Conceptual- 
ism. So is it, again, in the case of memory. 
The act is usually described as if there were 
present in the mind the two elements of a pres- 
ent mental representation and an intuition, or 
presentation, its prototype, to' which it is re- 
ferred ; while the picture before the mind is but 
one, as an examination of consciousness during 
the act will very readily show. And the same 
truth is illustrated in innumerable ways by 
the debates and disputes of philosophers, past 
and present, as to mind and the faculties in 
which it is manifested. As a typical instance 
may be given the analysis of consciousness left 
us by Mr. John Stuart Mill as compared 
with that presented in Sir William Hamilton's 
Lectures. "Was not each describing what was 
present in his own mind? Whence the discre- 
pancy ? 

Moreover, a clear apprehension of the con- 
stituent elements of a mental state does not 



16 THE CONCEPTION OF THE INFINITE. 

always seem to be necessary, from a practical 
point of view, to enable one to use that state 
as a unit, with substantial accuracy. The Nom- 
inalist, the Realist, and the Conceptualist all 
speak the same language, refer to the same 
objects, and, in the use of the complex mental 
phenomenon which they so variously analyze, 
are at one. A seaman, practised in the judg- 
ment of distances by a long training in his vo- 
cation, may be much more accurate in his judg- 
ments of distance than the accomplished author 
of the "New Theory of Vision" himself, though 
he may be quite ignorant of the mental process 
by which he arrives at his conclusions. The 
process itself is not directly affected by an ana- 
lytic consciousness of the steps of the process, 
nor is its result. The mental state is in most 
cases recognized only as a unit, since it is as a 
unit that it is useful to the individual ; and the 
name by which it is known expresses the gen- 
eral impression conveyed to the mind when the 
state is called up. 

Now, it is manifest, that when one begins to 
analyze this vague and indefinitely grasped 
total, and to separate it into its elements, it is 
quite possible for him to confound some of the 



INTRODUCTORY. 17 

elements with elements somewhat similar, or 
to imagine the presence of elements closely 
connected by the laws of association with ele- 
ments really present; in short, to find what is 
not there, and what, when in practice he uses 
the word indicating the state as a unit, he never 
means to express by it. And, in view of this 
fact, we may see how it is possible for the curi- 
ous error regarding the inconceivability of the 
infinite, which has been adverted to, to have 
arisen, and to have held its place in the writings 
of philosophers. The word has always been 
.used, and the ideas for which it stands have 
often been in men's minds ; but in attempting 
to explicate the conception, we find that almost 
all place among the qualities it connotes a 
notion drawn from finites, and which is contra- 
dictory to the essential character of the concep- 
tion. Few men have talked more about the 
infinite than Sir William Hamilton ; and it is 
probable that every time he used the word it 
called up very much the same mental state in 
his mind as that which arose in the mind of 
Mr. John Stuart Mill when he read Sir Wil- 
liam's Lectures ; but the conception thus called 
up certainly did not contain the warring ele- 

2* 



18 THE CONCEPTION OF THE INFINITE. 

ments which Sir William finds in it when he 
undertakes to prove it unthinkable. And when 
Mr. Mill criticises Sir William Hamilton, and 
declares the infinite not inconceivable, he prob- 
ably meant by the infinite just what Hamilton 
did ; and yet when he tries to prove its con- 
ceivability, he finds in it what was not really 
contained in the mind of either when the word 
was used. 

The problem is simply one of psychological 
analysis, — a question of what may be the true 
content of a complex mental state ; and the 
fact that such an analysis may be incorrectly 
made need not surprise us, since erroneous an- 
alyses are only too common. What may well 
create surprise, however, is that the difficulties 
into which this erroneous analysis has led those 
guilty of it, the antinomies of which it is the 
source, have not called attention to the error, 
and that a rigorous analysis has not been em- 
ployed to eliminate it. I will consider in the 
remaining chapters of this little book the 
errors which have arisen from a mistaken no- 
tion of the content of our conception of the 
infinite, and show that they all arise from a 
foreign and contradictory element inadvertently 



INTR OD UCTOR F. 19 

admitted as part of the conception, and that, 
this element being eliminated, the conception 
is in no respect inconceivable, nor does it pre- 
sent any difficulties not presented by any other 
concept or general notion. 






CHAPTEB II. 

THE CONCEPTION NOT QUANTITATIVE. 

We will suppose two parallel straight lines, 
A and B, unlimited in extent, and intersected 
by perpendiculars, ab, a'b', a"b", etc., drawn at 
equal distances from each other. 



b V b" 

It is evident that each division upon A is equal 

to its corresponding division upon B, and the 

sum of any number of divisions upon A will 

equal the sum of a similar number upon B. 

Since, therefore, each division upon the one line 

has its corresponding division upon the other," 

will not the equation hold .good when all the 

divisions are considered ? That is, will not the 

sum of all the divisions upon A be equal to the 

sum of all the divisions upon B ? And will not 

the sum of all the divisions on both lines be 

equal to twice the sum of all the divisions on 

either? Must we not here regard one infinite 

as greater than another ? 
20 



THE CONCEPTION NOT QUANTITATIVE. 21 

Much depends upon the answer to this ques- 
tion, as it will reveal very clearly the content 
which one attributes to his conception of the 
infinite. To the giving of the wrong answer 
may be traced that misconception of the true 
nature of infinity which has been such a fruitful 
source of supposed antinomies. Broadly stated, 
the question is, Can infinites be regarded as 
comparable with each other, as greater or less 
than, or equal to, each other ? Let us consider 
the case of the parallel lines. 

It is true that we must consider each divis- 
ion on the one line equal to each division on 
the other ; and taking any number of divisions 
on the one, and adding them to an equal num- 
ber of divisions on the other, we obtain a sum 
equal to twice the number of given divisions on 
either. But when we say " all the divisions on 
the one are equal to all the divisions on the 
other," we speak of the lines as quantitative 
wholes, and introduce an error with the word 
all. 

To conceive of a thing as a ivhole, we must 
assign to it limits. In saying " the whole" of 
any object, we refer to„ those limits beyond 
which there is none of that object. In re- 



22 THE CONCEPTION OF THE INFINITE. 

garding any object as a quantitative whole, we 
necessarily think it as finite. 

"When we compare one line with another, 
and declare its extent greater or less than that 
of the other, we mean that, when the one is 
applied to the other, its limits extend be- 
yond or fall within the limits of the other. 
In other words, we give the difference between 
the distances included within their respec- 
tive limits. Measuring is merely giving the 
distance between limits. "When two lines are 
infinite, we have no point to measure from, 
and no point to measure to, and no measure- 
ment — therefore no comparison — is possible. 
It is a palpable contradiction to compare (i.e., 
give relations of measurement between the re- 
spective limits of) two infinites (i.e., things 
which cannot be measured, as having no 
limits). 

The terms longer, shorter, and equal, can, there- 
fore, have no meaning as aj?plied to infinite lines. 
They can be used only in speaking of the finite. 
We cannot, then, say that one infinite is greater 
or less than another, and just as little can we 
say that all infinites are equal; for any such 
proposition, however possible in words, is im- 



THE CONCEPTION NOT QUANTITATIVE. 23 

possible in thought, and is an attempt to join 
contradictory notions. 

In such cases as the above, where the lines 
are nowhere limited, the impossibility of an in- 
crease in length may be clearly seen. A line 
can only be lengthened by. adding to it at its 
extremities, and it is impossible that a line with- 
out ends should be added to. If one holds that 
the sum of two such lines is greater than either 
line separately, he simply states that that may 
be increased, the very conception of which pre- 
cludes the possibility of its increase. 

There are cases, however, in which the error 
of a wrong conclusion is not so immediately pal- 
pable as in the case just stated; for example, the 
case of a line limited at but one point. Suppose 
the line AB limited only at the point A. 



Continue the line to C. If now the line be di- 
vided by points placed at equal distances from 
each other, into equal divisions, AC containing 
three such divisions, will not the whole line CB 
be greater by three divisions than the whole 
line AB ? 

AB is limited at A ; consequently there is 



24 THE CONCEPTION OF THE INFINITE. 

nothing to prevent our adding to it at its one 
extremity. Does it not seem natural to assume 
that in thus adding we increase the sum total of 
the line ? "We have gone through the same 
process as that by which we increase finite 
lines. 

When we recollect, however, that the line 
AB is limited only at one point, and is not, 
therefore, as a line, defined (for two points are 
necessary to define a line), the impossibility of 
regarding "it as a quantitative whole is evident, 
and the impossibility of increasing or diminish- 
ing its length, as a whole, necessarily follows. 
All of CB is. not greater than all of AB, be- 
cause the word all (in its quantitative sense*) 
cannot be applied to either. 

Suppose we attempt a comparison of the two 
lines. Let CB be superposed upon AB in 
such a manner that C will fall upon A. The 
two lines will then have the one limit in com- 
mon ; but one limit does not furnish data for 
lineal comparison, and no judgment can be 
formed as to the comparative length of the 
two lines. Where the one line is regarded as 

* This distinction will be noticed later. 



THE CONCEPTION NOT QUANTITATIVE. 25 

greater than the other, from the fact that three 
of its divisions project beyond the limit of the 
other, the measurement begins with an im- 
agined point infinity, which is regarded as a 
common limit of the other extremities of the 
two lines, and concludes with the limits at A 
and C. 

The error of such an attempt at measurement 
is clearly revealed by beginning to measure at 
A and C. It may be here justly remarked 
that we have before us a concrete instance of 
the truth of the old adage, that is a poor (meas- 
uring) rule that will not work both ways. 
The illusion disappears when Ave begin to 
measure at the given and only limits. 

Now, drawing the necessary inference from 
the foregoing, we may answer the question 
whether a line altogether without limits is not 
greater than a line limited at but one point, by 
saying that the very nature of the conceptions 
precludes the possibility of the words greater or 
less being applied to either ; that neither of the 
lines can be regarded as a quantitative whole ; 
and that, consequently, the question is a mean- 
ingless one. 

When we turn our attention to the considera- 
3 



26 THE CONCEPTION OF THE INFINITE. 

tion of surfaces, we meet with similar misappre- 
hensions, and arising from the same cause. It 
is asserted, for example, that if we suppose three 
parallel straight lines infinite in extent, one of 
which, A, is separated by a distance of two 
metres from the middle line B, while the other, 
C, is distant from it but one metre, 



we must conclude that the surface included be- 
tween A and B is double the surface included 
between B and C. 

Let the lines be intersected by perpendiculars 
one metre apart. Do we not find that each 
metre in the surface contained between B and 
C has corresponding to it two metres in the 
other surface ? And is not this proportion the 
same for twenty or for two hundred as for one, 
and quite independent of number? If, then, the 
proportion will hold good for any number, why 
will it not hold good when all the divisions are 
considered ? 

Or if we consider the angle formed by the 



THE CONCEPTION NOT QUANTITATIVE. 27 

intersection of two infinite straight lines, must 
we not conclude that increasing the angle will 
increase the area of the surface included be- 
tween the lines forming the angle ? And that 
diminishing the angle will diminish the area of 
the included surface ? Let A and B be two 
infinite straight lines intersecting at C ; 




and let A'C be an infinite straight line, making 
a smaller angle with CB than is made by AC. 
Must we not affirm that the surface ACB is 
greater than the surface A'CB, and that it is 
equal to the sum of A'CB and ACA' ? The 
answer to be given in these cases is evidently 
the answer which has already been given. The 
quantitative relations of equality and inequality 
certainly hold good for all quantities; and 
taking the case of the surfaces included be- 
tween the parallel lines, we must admit that any 
number of the divisions between A and B will 



28 THE CONCEPTION OF THE INFINITE. 

have double the area of a similar number of the 
divisions between B and C ; but when we speak 
of all the divisions, we do not refer to any 
number; we do not express by the word a 
quantity; and where the notion of quantity is 
wanting, manifestly, quantitative relations will 
not hold. So in the case of the increased or 
diminished angle. 

It is unnecessary to multiply instances, as the 
principle is in all cases the same. In general,, 
wherever the limit is removed in any one direc- 
tion, whether in the case of lines, of surfaces, or 
of solids, the object can no longer be regarded 
as a quantitative whole, and is not to be con- 
sidered finite. 

It remains to consider a class of cases of an 
apparently different nature. It is argued, for 
example, that an infinite series of dollars will 
exceed in value an infinite series of cents ; that, 
where the unit differs, the difference will extend 
to the series in its totality. It is easy to show 
the error of such a position by showing what 
the assertion necessarily involves. 

Suppose that, instead of counting one cent in 
the one series to each dollar in the other, we 
vary our mode of procedure by counting one 



THE CONCEPTION NOT QUANTITATIVE. 29 

hundred cents in the one to each dollar in the 
other. . It is true that the one series is exhausted 
one hundred times as rapidly as the other ; but 
since they are both infinite (will never end), we 
may continue this forever (to infinity), and the 
two series will have equal values. Or we may 
count two hundred cents in the one to each 
dollar in the other, or three hundred, or four 
hundred; so that the same series will be equal 
to, or twice, thrice, or four times as great as 
another ; its value depending merely on the 
mode of reckoning. 

If it is just to conclude that an infinite series 
of dollars is one hundred times as great in value 
as an infinite series of cents, we must also accept 
the conclusions arrived at by the other modes 
of reckoning; all are based on the same prin- 
ciple. Our only escape- from these warring con- 
clusions is to declare the principle underlying 
all the modes of reckoning an erroneous one. 
The error lies in regarding these infinite series 
as in any way capable of being compared with 
each other; in looking upon them as quanti- 
tative wholes. 

It is clear, therefore, that the true conception 
of the infinite is not quantitative but qualitative, a 

3* 



30 THE CONCEPTION OF THE INFINITE. 

fact which has been very generally overlooked, 
and with disastrous consequences much the same 
in all cases. 

I have said that the word " whole," as pre- 
dicating totality, cannot be applied to infinites, 
as naturally follows from the qualitative nature 
of our conception of the infinite; but the use 
of the word is really unavoidable, and, when 
it is used, it should be borne in mind that, ap- 
plied to infinites, the word has a certain quali- 
tative sense quite different from that in which 
it is applied to finites. If I were to speak of 
" all (possible) men" as distinguished from 
" all (possible) angels," I should in no respect 
limit the infinite possible number of either 
(the phrase " infinite number," though, strictly 
considered, incorrect, may be understood in a 
qualitative sense as expressing unlimited units), 
and my conceptions would be not quanti- 
tative but qualitative ; for the mind would be 
occupied, not with the number of objects, but 
with certain conditions which any object must 
satisfy to fall within the one class or the 
other. 

If, on the other hand, I speak of " all (actual) 
men," I may regard them as a definite known 



THE CONCEPTION NOT QUANTITATIVE. 31 

or unknown number, and my conception may 
be quantitative. 

Similarly, if, in speaking of an infinite line, I 
should say, " the line AB is in all its parts a 
straight line," I could only mean that one of 
the general conditions of the line is straight- 
ness, and that, whatever part of it may be 
thought of, it must agree with this condition. 
The " all" is in such a case equivalent to "any," 
not. to "every," and the two meanings may 
easily be confounded. Indeed, there is some- 
thing misleading in the very expression, " an 
infinite line," for the unreflective mind is apt 
to regard the object to which it is applied as a 
unit, a whole ; it is very necessary in using the 
phrase to keep in mind its true meaning. 

It may be objected to what I have here 
brought forward that any theory which denies 
that we have knowledge of the infinite as a 
whole may justly be called agnostic. If we do 
not know the infinite as a whole, do we not 
know only its parts, which are finite ? And 
have we any true knowledge of the infinite at 
all? 

I answer, the conception of a part is, as well 
as the conception of a whole, quantitative, and 



32 THE CONCEPTION OF THE INFINITE. 

an object recognized as part of a greater object 
is thereby necessarily recognized as finite, But 
if the object before the mind is not quantita- 
tively regarded at all, either as whole or part, 
our conception may be of the infinite. The 
plausibility of the objection arises from its con- 
founding two very different things, the distinc- 
tion between which will be more clearly drawn 
in the last chapter of this monograph. 

But as a preliminary answer to the objection, 
I may say that the assertion that we do not 
know the infinite as a whole is by no means 
equivalent to the assertion that we do not know 
the infinite. We do not know the moon as 
square, but that would scarcely prove that we 
have no knowledge of the moon, since the 
notion of squareness forms no part of a true 
knowledge of that object. Just as little is the 
quantitative conception of totality necessary to 
a knowledge of the infinite. 

It is not agnosticism to declare the mind 
unable to think that which is in its nature self- 
contradictory, — to define an object as infinite 
and then think it as limited ; while, on the 
other hand, any theory which maintains that 
we may know as a whole that which, in its very 



THE CONCEPTION NOT QUANTITATIVE. 33 

conception, precludes the possibility of its being 
so considered, may be accused of the direst 
agnosticism, as discrediting a fundamental law 
of thought, the law of non-contradiction. The 
theory attacked may as a last resource avail 
itself of the old argumentum ad hominem, and re- 
mark in pointed terms that the kettle is not as 
black as some other vessels in the speculative 
kitchen. 



CHAPTER III. 

THE ANTINOMIES OF HAMILTON. 

The evils resulting from overlooking the fact 
that the conception of the infinite is qualitative 
are evident when we examine some of the rea- 
sonings based upon the supposition that the 
conception contains a quantitative element. 

And first I will examine, so far as it touches 
the point in question, that agnostic theory de- 
veloped by Sir William Hamilton under the 
name of the Philosophy of the Conditioned, 
the fundamental principle of which is that " all 
that is conceivable in thought lies between 
two extremes, which, as contradictory of each 
other, cannot both be true, but of which, as 
mutual contradictories, one must."* The rea- 
soning in which Sir William applies this prin- 
ciple to our knowledge of space is worthy of 
attention. "... We conceive space," he says, 
— " we cannot but conceive space. I admit, 
therefore, that Space, indefinitely, is a positive 



* " Metaphysics," New York, 1880, p. 527. 
34 






TEE ANTINOMIES OF HAMILTON. 35 

and necessary form_ of thought. But when 
philosophers convert the fact, that we cannot 
but think space, or, to express it differently, 
that we are unable to imagine anything out of 
space, — when philosophers, I say, convert this 
fact with the assertion that we have a notion, 
a positive notion, of absolute or of infinite space, 
they assume not only what is not contained in 
the phenomenon, nay, they assume what is 
the very reverse of what the phenomenon man- 
ifests. It is plain that space must either be 
bounded or not bounded.. These are contra- 
dictory alternatives; on the principle of Con- 
tradiction they cannot both be true, and, on 
the principle of Excluded Middle, one must 
be true. This cannot be denied without de- 
nying the primary laws of intelligence. But 
though space must be admitted to be neces- 
sarily either finite or infinite, we are able to 
conceive the possibility neither of its finitude 
nor of its infinity. 

" We are altogether unable to conceive space 
as bounded, — as finite ; that is, as a whole, be- 
yond which there is no further space. Every 
one is conscious that this is impossible. It con- 
tradicts also the supposition of space as a neces- 



36 THE CONCEPTION OF THE INFINITE. 

sary notion ; for if we could imagine space as 
a terminated sphere, and that sphere not itself 
enclosed in a surrounding space, we should not 
be obliged to think everything in space; and, 
on the contrary, if we did imagine this termi- 
nated sphere as itself in space, in that case we 
should not have actually conceived all space as 
a bounded whole. The one contradictory is 
thus found inconceivable ; we cannot conceive 
space as positively limited. 

" On the other hand, we are equally power- 
less to realize in thought the possibility of the 
opposite contradictory; we cannot conceive space 
as infinite, as without limits. You may launch 
out in thought beyond the solar walk, you may 
transcend in fancy even the universe of matter, 
and rise from sphere to sphere in the region of 
empty space, until imagination sinks exhausted; 
with all this, what have you done ? You have 
never gone beyond the finite ; you have attained, 
at best, only to the indefinite, and the indefinite, 
however expanded, is still always the finite. . . . 
Now, then, both contradictories are equally in- 
conceivable, and, could we limit our attention to 
one alone, we should deem it at once impossible 
and absurd, and suppose its unknown opposite 



THE ANTINOMIES OF HAMILTON. 37 

as necessarily true. But as we not only can, but 
are, constrained to consider both, we find that 
both are equally incomprehensible; and yet, 
though unable to view either as possible, we are 
forced by a higher law to admit that one, but 
one only, is necessary." 

It is evident that the difficulties in which Sir 
William has here involved himself are gratu- 
itous. The argument used to prove the latter 
of the contradictories inconceivable breaks down 
upon a careful examination. 

"We may indeed " rise from sphere to sphere 
in the region of empty space" without trans- 
cending the finite, and the attempt to thus 
transcend it is as hopeless as would be the at- 
tempt of the peacock to escape from his feet by 
frying; for we cannot arrive at the unlimited 
while we carry our limits^ with us. Each suc- 
cessive stage simply places the limits farther 
apart, and in no respect helps us to do away 
with them altogether. 

Such a mode of procedure forcibly reminds 
one of the amusing person in Chamisso's poem, 
who supposed that by quickly turning himself 
around he could cause his queue to hang in 
front : 



38 THE CONCEPTION OF THE INFINITE. 

" Er dreht sich links, er dreht sich rechts, 
Es thut nichts Gut's, es thut nichts Schlecht's, 
Der Zopf, der hiingt ihm hinten." 

And how analogous is the condition of one 
who thinks that the way to reachjthe infinite is 
to endlessly continue this hopeless journey be- 
yond "the universe of matter" to' that of the 
hero as portrayed in the last verse : i 

" Und seht, er dreht sich immer noch 
Und denkt : es hilft am Ende doch— 
Der Zopf, der hangt ihm hinten." 

It is not by adding space to space that we 
arrive at the idea of infinite space, and imagi- 
nation may well sink exhausted in the attempt 
to find the end of the endless. 

But to think space as infinite it is by no 
means necessary to take this journey; and, so 
far from proving that we cannot regard our 
notion of space as infinite, the failure of any 
such attempt to know it as a whole is the surest 
evidence that it is indeed infinite. The latter 
of the contradictories is thus found to be incon- 
ceivable only when we suppose a quantitative 
element in our conception of infinity, and, this 
error corrected, the antinomy disappears. 



THE ANTINOMIES OF HAMILTON. 39 

Sir William has applied the Law of the Con- 
ditioned also to the minimum of space: "That 
the conceivable," he continues, " lies always be- 
tween two inconceivable extremes is illustrated 
by every other relation of thought. We have 
found the maximum of space incomprehensible; 
can we comprehend its minimum? This is 
equally impossible. Here, likewise, we recoil 
from one inconceivable contradictory only to 
infringe upon another. Let us take a portion 
of space, however small ; we can never conceive 
it as the smallest. It is necessarily extended, 
and may, consequently, be divided into a half or 
quarters, and each of these halves or quarters 
may again be divided into other halves or quar- 
ters, and this ad infinitum. But if we are unable 
to construe to our mind the possibility of an 
absolute minimum of space, we can as little 
represent to ourselves the possibility of an in- 
finite divisibility of any extended entity." 

This is at bottom a repetition of the above- 
mentioned error. Whether we regard space, 
with the Kantian, as in its nature wholly com- 
posite, and always capable of further subdivision ; 
or, with the Berkeleyan, as composed of minima 
visibilia, themselves not admitting; of subdi- 



40 THE CONCEPTION OF THE INFINITE. 

vision ; in neither case are we forced into a 
choice of two inconceivables. The Berkeleyan 
would claim that the difficulty of conceiving a 
component part of any extended entity as itself 
non-extended arises from the fact that the min- 
imum visibile is represented by the imagination 
as extended, the notion of extension being car- 
ried over from those objects to which it right- 
fully belongs, and is, consequently, not a true 
minimum visibile. "While the Kantian may main- 
tain that there is nothing inconceivable in an 
infinite series, rightly understood. If we sup- 
pose a series to be infinite, we cannot, of course, 
represent it to the mind as a completed whole, 
but that is unnecessary to — it is incompatible 
with — our recognition of it as infinite. The an- 
tinomy, like its predecessor, disappears as soon 
as it is recognized that there is no quantitative 
element in our conception of the infinite.* 

* It is odd that the statement that we are unable to con- 
ceive of any portion of space as the smallest possible, and 
not itself divisible into spaces, should be so constantly 
allowed to pass unchallenged. The question is not whether, 
when we have carried our subdivision of a surface so far 
that an apparently unextended point of color alone is left, 
we can in imagination substitute for that point an extended 



THE ANTINOMIES OF HAMILTON. 41 

Let us turn now to Sir William's application 
of the law to our conception of time: 

"In like manner Time; — this is a notion 
even more universal than space, for while we 
exempt from occupying space the energies of 
mind, we are unable to conceive these as 
not occupying time. Thus, we think every- 
thing, mental and material, as in time, and out 
of time we can think nothing. But if we at- 
tempt to comprehend time, either in whole or in 

surface, and proceed to subdivide that, convinced that any 
system of relations developed from the latter may lawfully 
be carried over to all possible future experiences of a simi- 
lar nature which may be found to be connected with the 
former ; but the question really is whether we can conceive 
that identical apparently unextended spot of color divided 
and subdivided. If we say that we are dividing it in 
thought, when our division proceeds thus through a rep- 
resentative, if we insist in applying to the object in the 
two cases the word "same," we should never forget that 
we are using this highly ambiguous word "same" in by no 
means the strictest of its four distinctly different senses, 
and should .be very sure that we are not practising self- 
deception by juggling with the several meanings of the 
word. When I say that with the Kantian hypothesis we 
are not forced to embrace what is inconceivable, I refer 
only to the point under consideration, the conceivability of 
series per se. 

4* 



42 THE CONCEPTION OF THE INFINITE. 

part, we find that thought is hedged in between 
two incomprehensibles. Let us try the whole. 
And here let us look back, — let us consider time 
a parte ante. And here we may surely flatter 
ourselves that we shall be able to conceive time 
as a whole, for here we have the past period 
bounded by the present; the past cannot, there- 
fore, be infinite or eternal, for a bounded infi- 
nite is a contradiction. But we shall deceive 
ourselves. We are altogether unable to con- 
ceive time as commencing ; we can easily rep- 
resent to ourselves time under any relative 
limitation of commencement and termination, 
but we are conscious to ourselves of nothing 
more clearly, than that it would be equally 
possible to think without thought, as to con- 
strue to the mind an absolute commencement 
or an absolute termination of time, — that is, a 
beginning and an end beyond which time is 
conceived as non-existent. Goad imagination 
to the utmost, it still sinks paralyzed within 
the bounds of time, and time survives as the 
condition of the thought itself in which we 
annihilate the universe. On the other hand, 
the concept of past time as without limit — with- 
out commencement — is equally impossible. We 



THE ANTINOMIES OF HAMILTON. 43 

cannot conceive the infinite regress of time; 
for such a notion could only be realized by the 
infinite addition in thought of finite times, and 
such an addition would itself require an eternity 
for its accomplishment. If we dream of effect- 
ing this, we only deceive ourselves by substi- 
tuting the indefinite for the infinite, than which 
no two notions can be more opposed. 

" The negation of a commencement of time 
involves, likewise, the affirmation that an infinite 
time has, at every moment, already run ; that is, 
it implies the contradiction that an infinite has 
been completed. For the same reasons we are 
unable to conceive an infinite progress of time; 
while the infinite regress and the infinite prog- 
ress, taken together, involve the triple contra- 
diction of an infinite concluded, of an infinite 
commencing, and of two infinities not exclusive 
of each other." * 

The statement that past time cannot be re- 
garded as infinite because limited by the present 

* " Metaphysics," New York, 1880, p. 529. See, also, 
Spencer's "First Principles," New York, 1875, p. 81: "How 
self-destructive is the assumption of two or more Infinites 
is manifest on remembering that such Infinites, by limiting 
each other, would become finite." 



44 THE CONCEPTION OF THE INFINITE. 

is based upon the erroneous supposition that 
what is limited at one point cannot be infinite. 
But, as has been shown in the preceding chap- 
ter, one point is not sufficient to define a line 
as finite; and time, which we represent to our- 
selves under the form of a continuous line, must 
be regarded as infinite unless limited at two 
points. Time past and time future are two in- 
finites, and as such are perfectly conceivable. 
The difficulty respecting the possibility of two 
infinites mutually exclusive of each other is a 
difficulty only under a false conception of the 
infinites as quantitative wholes, and may easily 
be made to disappear. 

The assertion, also, that the past cannot be 
infinite, as "a bounded infinite is a contradic- 
tion," may well be scanned. Arguments drawn 
from the etymological signification of a word 
are of small value, unless that expresses the 
true and whole content of the word. That such 
is not the case here is evident. A line limited 
at but one point is certainly not finite, for it 
cannot be regarded as a quantitative whole : 
cannot be increased, diminished, or compared 
in length with other lines ; in short, is not sub- 
ject to the conditions of the finite. If, then, for 



THE ANTINOMIES OF HAMILTON. 45 

etymological reasons, we exclude it from the 
class of infinites, we have the finite, the infinite, 
and a third class, a tertium quid, which lies be- 
tween the two, and might be humorously de- 
scribed as " infinite at one end." But, etymol- 
ogy aside, there is no difficulty in classing such 
a line with one that has no limits, for they are 
subject to precisely the same conditions, and 
equally distinct from the finite. However, the 
appellation is a matter of taste ; the thing which 
it is important to bear in mind is that a line 
with but one limit can no more be regarded as 
a quantitative whole than a line absolutely with- 
out limits ; and, whether we choose to call time 
past infinite or finite, that we may have a clear 
knowledge of it as limited only by the present, 
without attempting to pass over its parts in 
succession, and thus arrive at the whole. 

The assertion, too, that the negation of a 
commencement of time " implies the contradic- 
tion that an infinite has been completed" is 
misleading. The word " completed" is an un- 
fortunate one to use in this connection, as it 
suggests to the mind the idea of progression 
from a beginning to an end. The denial of a 
commencement of time does imply that an in- 



46 THE CONCEPTION OF THE INFINITE. 

finite is past, but not that it is completed in any 
such sense as to be enclosed within limits, for 
it is quite conceivable that it never began, and 
the present moment is, by supposition, the only 
limit. 

Removing the misconceptions just noticed, the 
whole force of the argument for the inconceiv- 
ability of time as infinite lies, as in the former 
cases, in the idea that the infinite may only be 
known by exhausting it in its totality, through 
the successive addition of its finite parts, and 
this antinomy also proves to be a gratuitous 
one. 

Having argued thus far the inconceivability 
of time as a maximum, Sir William turns, as in 
his discussion on our knowledge of space, to a 
consideration of time as a minimum: 

" ~Noxv take the parts of time, — a moment, for 
instance; this we must conceive as either divis- 
ible to infinity, or that it is made up of certain 
absolutely smallest parts. One or other of these . 
contradictories must be the case. But each is, 
to us, equally inconceivable. Time is a pro- 
tensive quantity, and, consequently, any part of 
it, however small, cannot, without a contradic- 
tion, be imagined as not divisible into parts, and 



THE ANTINOMIES OF HAMILTON. 47 

these parts into others ad infinitum,. But the 
opposite alternative is equally impossible; we 
cannot think this infinite division. One is neces- 
sarily true; but neither can be conceived pos- 
sible. It is on the inability of the mind to con- 
ceive either the ultimate indivisibility or the 
endless divisibility of space and time that the 
arguments of the Eleatic Zeno against the pos- 
sibility of motion are founded, — arguments 
which at least show that motion, however cer- 
tain as a fact, cannot be conceived possible, as 
it involves a contradiction." * 

With reference to the former of the alterna- 
tives offered, the Berkeleyan would answer that 
in the case of " protensive" quantity, as in the 
case of extensive, the difficulty of conceiving 
the unit, itself indivisible, lies in the imagina- 
tion, and may, with precautions, be obviated; 
while with reference to the latter, the Kantian 
may answer, as before, that an infinite series is 
not intrinsically unthinkable. I may remark, 
en passant, that the idea of the theoretic impos- 
sibility of motion is a wholly erroneous one, and 
falls with the errors upon which it is based. 

* "Metaphysics," New York, 1880, pp. 529-30. 



48 THE CONCEPTION OF THE INFINITE. 

One turns from an examination of Sir Wil- 
liam Hamilton's application of the Law of the 
Conditioned to Space and Time with a convic- 
tion that if the Philosophy of the Conditioned 
has no better props to sustain it than these 
prove to be, it may turn out as insecure an 
edifice as the house that a certain foolish indi- 
vidual, according to the parable, founded upon 
sand. 

It remains to consider a case which ap- 
parently militates against the theory that an 
infinite series can never be regarded as a 
whole. 

Let us suppose a point moving uniformly 
along the line AB, over the whole of which it 
can pass in one minute. In J of a minute it 
will have passed over \ of the line; in \ of a 
minute over \ more ; in \ of a minute \ more, 
etc. When the minute is completed the point 
will have passed over the whole line. Has it 
not passed successively over the whole series, 
thus completing it, and arriving at as its 
lower limit? And may we not say that the 
sum of all the terms in the series is equal to 
the whole line passed over ? Is it not a quan- 
titative whole ? 






THE ANTINOMIES OF HAMILTON. 49 

A little reflection will reveal the fallacy in 
this reasoning. The series is not completed at 
all, but is truly infinite. It is limited at one 
point by the highest term (J), but is not limited 
at another point by a lowest term (0) ; for the 
can only be assumed as a limit to the series by 
breaking the law of the series, which is that 
each term shall be half as great as the one pre- 
ceding. 

We can never, by halving something, arrive 
at nothing; a division of substance will never 
give us that which is not substance. The 0, 
since it does not make one in the series, cannot 
limit the series. The Kantian may answer the 
question by reasoning as follows : " The point 
in question passes over the whole line, not by 
completing the descending series until it arrives 
at a lowest term in the simple, and from that 
passes to zero, but by the successive addition of 
spaces, which are themselves composites. ' As 
space is not a composite of substances (and not 
even of real accidents), if I abstract all compo- 
sition therein, nothing, not even a point, re- 
mains ; for a point is possible only as the limit 
of a space, consequently of a composite. Space 
and time, therefore, do not consist of simple 



50 THE CONCEPTION OF THE INFINITE. 

parts.' * We cannot, therefore, consider any 
member of the series in question as the smallest 
possible, nor the zero as a limit to the series; 
nor can we regard the series as in any sense 
completed. If we discontinue the subdivision 
at any point whatever, we may justly say that 
the foot or yard contains all the terms of the 
series. But when the point has reached zero it 
has reached it by breaking the series, not com- 
pleting it. A completion the law of the series 
renders impossible." f And the Berkeleyan 
might answer that a completion of the journey 
along the line by no means implies the com- 



*" Critique of PureEeason;" "Observations on the Sec- 
ond Antinomy." Ed. Bohn. 

f How the series can he broken by the progressive motion 
of a point over a line when, by hypothesis, the series is 
throughout applicable to the line (i.e., the line is infinitely 
divisible), the Kantian must explain; the fact remains that 
neither an infinite nor a finite series can be completed by 
breaking its law ; and unless we claim that half of some- 
thing, when that something is small enough, is equal to 
nothing, we cannot bring the zero into the series as a limit. 
Even could this be done, the desired point would not be 
established, as the series would then be limited at two points, 
consequently in no sense infinite, and its completion would 
not be the completion of an infinite series. 



THE ANTINOMIES OF HAMILTON. 51 

pletion of an infinite series, since relations of 
quantity may be divorced from all content of 
actual being, and used purely symbolically. 
Mathematical reasonings, lie would argue, are 
applicable to space only within the limits of a 
possible perception ; and the series may be truly 
infinite, though the line-portions to which some 
of its members are applied may be limited in 
number.* 

This problem, it will be seen, is simply the 
old puzzle of Achilles and the tortoise in a 
somewhat altered dress ; and the answer which 
has been given to that puzzle — that the series 
"runs into infinitesimals which are practical 
zeros, and, even if theoretically infinite in num- 
ber, really are all included in that finite length 
which Achilles will quickly get over"f — is not 

* I have taken up both positions to bring out clearly the 
fact that the assertion of the conceivability of the infinite 
is quite independent of metaphysical theories as to the nature 
of space and time. The two questions are wholly distinct, as 
I will show in a later chapter. 

f This is the solution of the problem given by the late Pro- 
fessor Atwater in his little book on the Elements of Logic. 
The explanation offered could not be much worse. "What, 
we may ask, is meant by a " practical zero," as distinguished 
from a theoretical? And the assertion that these practical 



52 THE CONCEPTION OF THE INFINITE. 

a true answer, since it regards a series, which, 
by the very terms of its statement, is incapable 
of completion, as a completed whole and equal 
to a finite. 

Before leaving this chapter I would remark 
that it is interesting to notice how wide-spread 
has been the conviction that the only way to 
arrive at a cognition of the infinite is to proceed 
to an endless addition of finites. Hamilton, we 
have seen, makes much of it. Mansel follows 
in his footsteps. Mr. Herbert Spencer quotes 
him with approbation. And Kant, as we shall 
see in the chapter following, did not escape the 
snare. " To have actually in the mind the idea 
of a space infinite," says Locke,* " is to suppose 
the mind already passed over, and actually to 
have a view of all those repeated ideas of space 
which an endless repetition can never totally 
represent to it; — which carries in it a plain con- 
tradiction." 

zeros, "even if theoretically infinite in number, really are 
all included" in a finite length, would seem to draw a dis- 
tinction between tbe theoretical and the practical, little to 
the advantage of the former. It makes theoretical about 
equivalent to unreal. 

* "Essay concerning Human Understanding," book ii. 
ch. xvii., \ 7. 



CHAPTER IV. 

KANT, MILL, AND CLIFFOKD. 

It is interesting to notice that the truth that 
our conception of infinity contains no quanti- 
tative element has been seen, like Thule, 
" through the mist" by several acute minds, 
who have yet not seen the truth with sufficient 
clearness to escape the common errors arising 
from the introduction of the contradictory ele- 
ment into their discussions. Immanuel Kant, 
although he has based the proof of the thesis of 
his first antinomy on a false conception of in- 
finity, and although, after correctly criticising 
the false conception, he himself lapses into it, 
yet perceived, and in so many words gave ex- 
pression to the fact, that the conception of the 
infinite is qualitative. 

The thesis of the first antinomy maintains 
that the world has a beginning in time, and is 
limited with regard to space, both of which 
propositions are denied in the antithesis. The 
proofs offered in support of the antithesis may 

5* 53 



54 THE CONCEPTION OF THE INFINITE. 

be passed over as extraneous to the subject ; 
those in support of the thesis I will quote, not 
for the purpose of again refuting them, for they 
are identical with those used by Sir William 
Hamilton in his antinomies, but that I may 
give the observations appended to them, which 
are very significant in their contextual connec- 
tion. The proof proceeds by assuming the 
truth of the antithesis, and then showing it to 
be impossible. 

" Granted, that the world has no begin- 
ning in time ; up to every given moment of 
time an eternity must have elapsed, and there- 
with passed away an infinite series of suc- 
cessive conditions or states of things in the 
world. Kow, the infinity of a series consists 
in the fact that it never can be completed 
by means of a successive synthesis. It fol- 
lows that an infinite series, already elapsed, 
is impossible, and that consequently a begin- 
ning of the world is a necessary condition of 
its existence. And this was the first thing to 
be proved. 

" As regards the second, let us take the oppo- 
site for granted. In this case the world must 
be an infinite given total of coexistent things. 



KANT, MILL, AND CLIFFORD. 55 

Now, we cannot cogitate the dimensions of a 
quantity, which is not given within certain lim- 
its of an intuition, in any other way than by 
means of the synthesis of its parts,* and the 
total of such a quantity only by means of a 
completed synthesis or the repeated addition of 
unity to itself. Accordingly, to cogitate the 
world, which fills all spaces, as a whole, the 
successive synthesis of the parts of an infinite 
world must be looked upon as completed, — that 
is to say, an infinite time must be regarded as 
having elapsed in the enumeration of all co- 
existing things, which is impossible. For this 
reason an infinite aggregate of actual things 
cannot be considered as a given whole, conse- 
quently, not as a contemporaneously given 
whole. The world is consequently, as regards 
extension in space, not infinite, but enclosed in 



* Kant says, in a foot-note, " We may consider an unde- 
termined quantity as a whole when it is enclosed within 
limits, although we cannot construct or ascertain its totality 
by measurement, — that is, by the successive synthesis of its 
parts. For its limits of themselves determine its complete- 
ness as a whole." This method being absent in the case of 
infinites, Kant thinks he can cognize them only by falling 
back upon the successive synthesis. 



56 THE CONCEPTION OF THE INFINITE. 

limits. And this was the second thing to be 
proved." * 

It will be noticed that in the first part of the 
proof the word completed (vollendet) is used in 
the manner before objected to as misleading. 
"When we speak of a series as " completed by 
means of a successive synthesis," we are apt 
to regard it as a whole, with a beginning as 
well as an end. Of course, when we are con- 
sidering time past as limited by the present, 
such an association must be unfortunate. 

The observations on the thesis are the fol- 
lowing : 

" In bringing forward these conflicting argu- 
ments, I have not been on the search for soph- 
isms for the purpose of availing myself of 
special pleading, which takes advantage of the 
carelessness of the opposite party, appeals to a 
misunderstood statute, and erects its unright- 
eous claims upon an unfair interpretation. Both 
proofs originate fairly from the nature of the 
case, and the advantage presented by the mis- 

* Immanuel Kant's " Sammtliche Werke," Leipzig, 1867. 
Dritter Band, s. 304. I have taken the rendering of Meikle- 
john's excellent translation of the Critique. 



KANT, MILL, AND CLIFFORD. 57 

takes of the dogmatists of both parties has been 
completely set aside. 

" The thesis might also have been unfairly 
demonstrated by the introduction of an erro- 
neous conception of the infinity of a given 
quantity. A quantity is infinite if a greater 
than itself cannot possibly exist. The quantity 
is measured by the number of given units — 
which are taken as a standard — contained in 
it. Now, no number can be the greatest, be- 
cause one or more units can always be added. 
It follows that an infinite given quantity, conse- 
quently an infinite world (both as regards time 
and extension), is impossible. It is, therefore, 
limited in both respects. In this manner I 
might have conducted my proof; but the con- 
ception given in it does not agree with the true 
conception of an infinite whole. In this there 
is no representation of its quantity ; it is not 
said how large it is ; consequently its concep- 
tion is not the conception of a maximum. "We 
cogitate in it merely its relation to an arbitra- 
rily assumed unit, in relation to which it is 
greater than any number. Now, just as the 
unit which is taken is greater or smaller, the 
infinite will be greater or smaller; but the in- 



58 THE CONCEPTION OF THE INFINITE. 

finity, which consists merely in the relation to 
this given unit, must remain always the same, 
although the absolute quantity of the whole is 
not thereby cognized. 

" The true (transcendental) conception of in- 
finity is that the successive synthesis of unity in 
the measurement of a given quantum can never 
be completed. Hence it follows, without possi- 
bility of mistake, that an eternity of actual suc- 
cessive states up to a given (the present) mo- 
ment cannot have elapsed, and that the world 
must therefore have a beginning. 

" In regard to the second part of the thesis, 
the difficulty as to an infinite and yet elapsed 
series disappears ; for the manifold of a world 
infinite in extension is contemporaneously given. 
But, in order to cogitate the total of this mani- 
fold, as we cannot have the aid of limits consti- 
tuting by themselves this total in intuition, we 
are obliged to give some account of our concep- 
tion, which in this case cannot proceed from 
the whole to the determined quantity of the 
parts, but must demonstrate the possibility of a 
whole by means of a successive synthesis of the 
parts. But as this synthesis must constitute a 
series that cannot be completed, it is impossible 



KANT, MILL, AND CLIFFORD. 59 

for us to cogitate prior to it, and consequently 
not by means of it, a totality. For the concep- 
tion of totality itself is in the present case the 
representation of a completed synthesis of the 
parts; and this completion, and consequently its 
conception, is impossible." 

We here find brought forward a conception 
of the infinite which is declared faulty; a dec- 
laration of the point in which it differs from the 
true conception; and a statement of what, ac- 
cording to Kant, the true conception really is. 
The false conception is that " a quantity is in- 
finite if a greater than itself cannot possibly 
exist." We may readily see that such a con- 
ception gives us, not an infinite, but a finite. 
Not only is the word " greater" inapplicable to 
infinites, but the very expression, " a quantity is 
infinite," is absurd, as involving a flat contra- 
diction. Kant was too keen a thinker not to 
see that that which admits of an addition ot 
units, and, consequently, of increase as a whole, 
cannot be infinite. This does not agree, he says, 
with the true conception of the infinite, in which 
"there is no representation of its quantity; it is 
not said how large it is ; consequently its con- 
ception is not the conception of a maximum." 



(JO THE CONCEPTION OF THE INFINITE. 

Could there be a clearer recognition of the fact 
that the conception is not quantitative ? 

But it is evident that Kant did not see the 
full force and the logical consequences of the 
statement. In the sentence immediately pre- 
ceding the one in which he recognizes the quali- 
tative character of the conception he uses the 
phrase "an infinite whole;" and in the sentences 
immediately following he brings forward a con- 
ception faulty in precisely the same respect as the 
one criticised: "We cogitate in it merely its rela- 
tion to an arbitrarily assumed unit, in relation to 
which it is greater than any number. Now, just 
as the unit which is taken is greater or smaller, 
the infinite will be greater or smaller; but the 
infinity, which consists merely in the relation to 
this given unit, must remain always the same, 
although the absolute quantity of the whole is 
not thereby cognized." That is, if we designate 
the infinite by a, the unit by b, and the infinity 
(the relation between a and b) by x, we find that 
a varies as b, and x remains always the same (this 
can only mean numerically the same). In this 
case x is simply an indefinite number, and the 
" absolute quantity of the whole" can certainly 
be cognized. When we say the infinity remains 



KANT, MILL, AND CLIFFORD. (jl 

always the same, the question naturally arises, 
The same in what? In amount? If so, we 
have but the finite. The error is here, perhaps, 
not quite so palpable, but is just as real as in 
the case which Kant criticises, and it is of pre- 
cisely the same nature. Both parts of the proof 
given in support of the thesis of course fall to 
the ground when this error is rectified. 

The last two observations are merely a re- 
statement of the proofs of the thesis. The re- 
mark made in the last one, that " in order to 
cogitate the total of this manifold, as we cannot 
have the aid of limits constituting by themselves 
this total in intuition, we are obliged to give 
some account of our conception, which in this 
case cannot proceed from the whole to the de- 
termined quantity of the parts, but must dem- 
onstrate the possibility of a whole by means of 
a successive synthesis of the parts," will lose 
all its force if the words " total" and " whole" 
are abstracted. If this manifold is to be known 
as a whole, we cannot, of course, arrive at a 
knowledge of it without cogitating all its parts 
as contained in it; but since it is impossible 
that it should be thus known, we may " give 
some account of our conception" by simply 



62 THE CONCEPTION OF THE INFINITE. 

stating that it is without limit, thus recognizing 
it as truly infinite. 

It seems odd that Kant should have seen an 
error, objected to it, and fallen into it on the 
same page; should have said that there is no 
representation of its quantity in the conception 
of the infinite, and then have called it a whole ; 
declared that the true conception does not say 
how large it is, and yet have affirmed that the 
infinity is always the same, while the series as a 
whole varies with the unit. But, with and in 
spite of all his inconsistencies, it must be allowed 
that he recognized the truth, however imper- 
fectly, that our conception of the infinite has, 
properly speaking, no quantitative element, but 
is purely qualitative. 

This truth has also been recognized, though 
less clearly, by John Stuart Mill, in his " Ex- 
amination of Sir William Hamilton's Philoso- 
phy," where he maintains, in opposition to Sir 
William, that the infinite is not inconceivable ; 
and we can see in the paragraphs which he 
devotes to the subject that Mill had in mind 
two distinct conceptions, a quantitative and a 
qualitative. To the former, which he calls the 
adequate conception, he acknowledges we cannot 



KANT, MILL, AND CLIFFORD. (33 

attain ; the latter, which is truly qualitative, 
though he does not apply to it that name, he 
claims to be a real conception, implying nothing 
inconceivable. 

His attempted refutation of Hamilton's argu- 
ment for the inconceivability of infinite space 
proceeds as follows :* 

" Our author goes on to repeat the argument 
used in his reply to Cousin, that Infinite Space 
is inconceivable, because all the conception we 
are able to form of it is negative, and a negative 
conception is the same as no conception. ' The 
infinite is conceived only by the thinking away 
of every character by which the finite was con- 
ceived.' To this assertion I oppose my former 
reply. Instead of thinking awa} 7 every char- 
acter of the finite, we think away only the idea 
of an end, or a boundary. Sir W. Hamilton's 
proposition is true of ' The Infinite,' the mean- 
ingless abstraction, but it is not true of Infinite 
Space. In trying to form a conception of that, 
we do not think away its positive characters. 
We leave to it the character of space ; all that 

* "Examination of Sir William Hamilton's Philosophy," 
Boston, 1868, vol. i., pp. 104 et seq. 



■ 



64 THE CONCEPTION OF THE INFINITE. 

belongs to it as space; its three dimensions, with 
all their geometrical properties. We leave to 
it also a character which belongs to it as Infi- 
nite, that of being greater than any other space. 
If an object which has these well-marked posi- 
tive attributes is unthinkable, because it has a 
negative attribute as well, the number of think- 
able objects must be remarkably small. Nearly 
all our positive conceptions which are at all 
complex include negative attributes. I do not 
mean merely the negatives which are implied in 
affirmatives, as in saying that snow is white we 
imply that it is not black, but independent neg- 
ative attributes superadded to these, and which 
are so real that they are often the essential char- 
acters, or differentiae, of classes. Our concep- 
tion of dumb is of something which cannot 
speak : of the brutes, as of creatures which have 
not reason ; of the mineral kingdom, as the part 
of nature which has not organization and life; 
of immortal, as that which never dies. Are all 
these examples of the Inconceivable ? So false 
is it that to think a thing under a negation is to 
think it as unthinkable. 

" In other passages, Sir W. Hamilton argues 
that we cannot conceive infinite space, because 



KANT, MILL, AND CLIFFORD. 65 

we would require infinite time to do it in. It 
would of course require infinite time to carry 
our thoughts in succession over every part of in- 
finite space. But on how many of our finite 
conceptions do we think it necessary to perform 
such an operation ? Let us try the doctrine 
upon a complex whole, short of infinite, such as 
the number 695,788. Sir W. Hamilton would 
not, I suppose, have maintained that this num- 
ber is inconceivable. How long did he think it 
would take to go over every separate unit of 
this whole, so as to obtain a perfect knowledge 
of that exact sum, as different from all other 
sums, either greater or less ? Would he have 
said that we could have no conception of the 
sum until this process had been gone through ? 
"We could not indeed have an adequate concep- 
tion. Accordingly, we never have an adequate 
conception of any real thing. But we have a 
real conception of an object if we conceive it by 
any of its attributes that are sufficient to distin- 
guish it from all other things. "We have a con- 
ception of any large number when we have 
conceived it by some one of its modes of com- 
position, such as that indicated by the position 
of its digits. We seldom get nearer than this 



(36 THE CONCEPTION OF THE INFINITE. 

to an adequate conception of any large number. 
But for all intellectual purposes this limited 
conception is sufficient; for it not only enables 
us to avoid confounding the number in our cal- 
culations with any other numerical whole, — 
even with those so nearly equal to it that no 
difference between them would be perceptible 
by sight or touch, unless the units were drawn 
up in a manner expressly adapted for displaying 
it, — but we can also, by means of this attribute 
of the number, ascertain and add to our concep- 
tion as many more of its properties as we please. 
If, then, we can obtain a real conception of a 
finite whole without going through all its com- 
ponent parts, why deny us a real conception of 
an infinite whole because to go through them all 
is impossible ? Not to mention that even in the 
case of the finite number, though the units com- 
posing it are limited, yet, Number being infinite, 
the possible modes of deriving any given num- 
ber from other numbers are numerically infinite ; 
and as all these are necessary parts of an ade- 
quate conception of any number, to render our 
conception even of this finite whole perfectly 
adequate would also require an infinite time. 
" But though our conception of infinite space 



KANT, MILL, AND CLIFFORD. (37 

can never be adequate, since we can never ex- 
haust its parts, the conception, as far as it 
goes, is a real conception. We completely 
realize in imagination the various attributes 
composing it. We realize it as Space. We 
realize it as greater than any given space. We 
even realize it as endless, in an intelligible 
manner, — that is, we clearly represent to our- 
selves that however much of space has been 
already explored, and however much more of 
it we may imagine ourselves to traverse, we are 
no nearer to the end of it than we were at 
first time ; however often we repeat the process 
of imagining distance extending in any direc- 
tion from us, that process is always susceptible 
of being carried farther. This conception is 
both real and perfectly definite. It is not vague 
and indeterminate as a merely negative notion 
is. We possess it as completely as we possess 
any of our clearest conceptions, and we can 
avail ourselves of it as well for ulterior mental 
operations. As regards the Extent of Space, 
therefore, Sir W. Hamilton does not seem to 
have made out his point; one of the two con- 
tradictor hypotheses is not inconceivable." 
One can see from these extracts how the 



68 THE CONCEPTION OF THE INFINITE. 

idea of quantity entered into and vitiated 
Mill's reasonings on the infinite. In arguing 
that our notion of infinite space is not a purely 
negative one, he enumerates several positive at- 
tributes that we leave to the conception when we 
take away the notion of limits. Among them 
he places "a character which belongs to it as 
Infinite, that of being greater than any other 
space." Evidently the character of being greater 
than any other space is a quantitative attribute, 
and can belong only to "a space," — a finite; 
while space infinite is not, properly speaking, " a 
space" at all. Since it has no size, it cannot be 
marked out from other spaces by its size. And 
Mill has manifestly committed the same error 
which has misled Sir William Hamilton, whom 
he criticises, when he says that it would, of 
course, require infinite time to carry our thoughts 
in succession over every part of infinite space. 
How can we speak of every part of that which 
is not quantitatively considered ? of that which, 
by its very definition, is incapable of being a 
whole? This is precisely Hamilton's error, and 
the cause of all his difficulties. A little farther 
on Mill speaks, just as Kant does, of "an infi- 
nite whole," never noticing the contradiction in 



KANT, MILL, AND CLIFFORD. 69 

the adjective; and in the paragraph following he 
repeats a former blunder in the statement that 
our knowledge of space can never be adequate, 
" since we can never exhaust its parts." 

It is strange that side by side with this con- 
ception of a quantitative infinite, which, to be 
adequately known, must be known as a whole, 
we should find a real, though imperfect, anal- 
ysis of the true conception, and an affirmation 
of its conceivability. We can conceive infinite 
space, says Mill ; we can conceive it as space, 
as greater than any given space, and even as 
endless, in an intelligible manner. When he 
comes to describe this intelligible manner of 
knowing the infinity of space, he uses an un- 
fortunate phrase, — " we clearly represent to our- 
selves that however much of space has been 
already explored, ... we are no nearer to 
the end of it than we were at first time ;" 
dragging in the idea of a limit, which is also 
done, as above remarked, by calling it greater 
than any other space. But with all these con- 
cessions to the old erroneous doctrine, it cannot 
be denied that Mill held that we may know In- 
finite Space in some other way than by a suc- 
cessive synthesis of finite spaces, and that he 



70 THE CONCEPTION OF THE INFINITE. 

attempted an enumeration of the psychical ele- 
ments comprehended by the conception, leaving 
out the notion of quantity. This conception, 
which he calls inadequate, but which he yet 
insists upon as a conception of the infinite, is 
qualitative, and harmonizes with the true char- 
acter of our conception of infinity. 

The last writer to whom I will advert as 
having had a knowledge, more or less clear, of 
the true qualitative nature of our conception 
of infinity, is Professor William Kingdon Clif- 
ford, by whose untimely death England has lost 
one of her acutest and most analytic minds. 
There are in his short paper entitled " Of Boun- 
daries in General," and devoted to the point, the 
line, and the surface, some interesting and sig- 
nificant passages, which I will quote, and upon 
which I will afterwards comment. The pas- 
sages are these : 

"Infinite ; it is a dreadful word, I know, until 
you find out that you are familiar with the 
thing which it expresses. In this place it 
means that between any two positions there 
is some intermediate position; between that 
and either of the others, again, there is some 
other intermediate ; and so on without any end. 



KANT, MILL, AND CLIFFORD. 71 

Infinite means without any end. If you went 
on with that work of counting forever, you 
would never get any farther than the begin- 
ning of it. At last you would have two po- 
sitions very close together, but not the same; 
and the whole process might be gone over 
again, beginning with those as many times as 
you like." . . . 

" In fact, when we said that there is an infi- 
nite number of points in a piece of line-room, 
we might have said a great deal more. Suppose, 
for instance, that any one said, ' How many 
miles is it possible to go up into space ?' The 
answer would of course be, ' An infinite num- 
ber of miles.' (Don't be frightened at this 
continual occurrence of the word infinite : it 
still means ' without any end,' and nothing 
more.) In this case, if you go a mile and count 
one, then another and count two, and so on, 
all we mean is that the process would never 
end. There would still be space left to go up 
into, however many millions of miles you had 
counted. But still all those miles would be 
counted and done with. Your task would 
have been distinctly begun, and there would be 
nothing more to say to the miles behind you. 



72 THE CONCEPTION OF THE INFINITE. 

But try now to count the points in a piece of 
line. You count one, two, three, four, a mil- 
lion points ; and your task is not even begun. 
The line is all there, exactly as it was before ; 
absolutely none of it is done with. The mil- 
lion points take up no more line-room than one 
point; that is to say, absolutely none at all. 
When, then, we are talking of the points in a 
piece of line, we must say not merely that there 
is a never-ending number of them (which there 
is), but that they are out of the reach of num- 
ber altogether. All the points in a line are 
not, properly speaking, a number of points at 
all. If we are going to speak about the num- 
ber of points in a line, we must settle before- 
hand that we are going to use the word in a 
new sense, which is not derived from counting, 
but from this very observation to which we 
have applied it. 

"Let us now make use of our idea of a 
path. When a point moves along a line, we 
know that between any two positions of it there 
is an infinite number (in this new sense) of in- 
termediate positions. That is because the mo- 
tion is continuous. Each of those positions is 
where the point was at some instant or other. 



KANT, MILL, AND CLIFFORD. 73 

Between the two end positions on the line, the 
point where the motion began and the point 
where it stopped, there is no point of the line 
which does not belong to that series. We have 
thus an infinite series of successive positions of 
a continuously moving point, and in that series 
are included all the points of a certain piece of 
line-room. May we say, then, that the line is 
made up of that infinite series of points ? 

" Yes, if we mean no more than that the 
series makes up the joints of the line. But no, 
if we mean that the line is made up of those 
points in the same way that it is made up of a 
great many very small pieces of line. A point 
is not to be regarded as a part of a line in any 
sense whatever." 

Evidently Clifford saw more clearly than either 
Kant or Mill that the notion of quantity is 
foreign to our conception of infinity. In speak- 
ing of the infinite extent of space, he does not 
make the absurd and tautological assertion that 
it would take an infinite time to exhaust it. He 
claims that when we call a thing infinite we 
know very well what it means : it means that it 
will never end. He never hints at any possi- 
bility of knowing the infinite as a whole, recog- 



74 THE CONCEPTION OF THE INFINITE. 

nizing very clearly that it cannot be a whole. 
And he has used the word number in two senses 
(which a careful reading of the extracts quoted 
will show to be a quantitative and a qualitative 
sense), to mark the difference between numbers 
regarded as constituent parts of wholes or sums 
total, and number regarded as unlimited units, 
having no relation to wholes of any sort, — a dis- 
tinction which precisely corresponds to the dis- 
tinction which I have drawn in a preceding 
chapter between the quantitative and the quali- 
tative uses of the word all, as applied to the 
members of a finite and an infinite series. 

But it is also evident that Clifford did not see 
the full significance and value of the truth which 
he recognized. This is clear from his contrast- 
ing the relation of linear miles to infinite exten- 
sion with the relation of mathematical points to 
a line, and indicating that in the latter case 
there is something peculiarly hopeless in the at- 
tempt to complete the line by the addition of 
such points. " You count one, two, three, four, 
a million points, and your task is not even 
begun. The line is all there, exactly as it was 
before ; absolutely none of it is done with." So 
that an addition of points, however long con- 



KANT, MILL, AND CLIFFORD. 75 

tinued, has no tendency to exhaust and complete 
the line. But in the case of infinite space, says 
Clifford, if you go a mile and count one, another 
and count two, and so on, all those miles would 
be counted and done with. " Your task would 
have been distinctly begun, and there would be 
nothing more to say to the miles behind you." 
But in reality, so far as they touch the question 
under discussion, the two cases are precisely 
similar. The points have not even begun to 
exhaust the line, because their addition has no 
tendency to exhaust it; their number has noth- 
ing to do with it.* And, similarly, one may 
justly hold that the miles counted have not even 
begun to exhaust the infinite space, since the 
increasing of the distance between limits has no 
tendency to make them approach the limits of 
that which is without limit ; since the adding of 
quantities has no tendency to make a sum equal to 
that which is not a quantity, and cannot be 
equal to anything ; and since adding mile to 
mile has no tendency to successively exhaust the 
parts of that which is not a whole, and can have 



* I speak here, of course, from Clifford's stand-point, 
assuming the infinite divisibility of a line. 



76 THE CONCEPTION OF THE INFINITE. 

no parts. The relation between a mile or a 
thousand miles and infinite extension is no 
closer than that which Clifford conceived be- 
tween a point or a thousand points and a given 
line. 

One cannot but see, therefore, that Clifford 
has not grasped the true nature of our concep- 
tion of infinity in all its consequences so thor- 
oughly as he might have grasped it. It will be 
seen, however, that he has discussed it more 
satisfactorily than either of the writers before 
cited. 



CHAPTER V. 

THE CONCEIVABLE AND THE EXISTENT. 

There are two quite distinct questions to be 
considered in a discussion of the infinite : the 
first is, "Can an infinite object be conceived?" 
and the second, "Does any infinite object really 
exist?" 

Now, one may answer the former of these ques- 
tions in the affirmative, and yet not be com- 
mitted to a similar answer to the latter. The 
two propositions, "An infinite object is conceiv- 
able" and "An infinite object exists" are dif- 
ferent in nature, and when the former is affirmed 
the latter still calls for proof. One may hold 
that space is subjective, a mere abstraction from 
experience of extended objects, and that, conse- 
quently, space, together with the imaginary line 
in space, exists only as it is produced in thought; 
and, though he may on that account deny that 
that line, as an imagined object, or any actually 
existent line is infinite, or that any line could 
possibly be made infinite, he may yet claim that 

he can conceive an infinite line. 

7* 77 



78 THE CONCEPTION OF THE INFINITE. 

If he belong to one school of thought he will 
not only claim that he can conceive space in- 
finite, but will assume on h priori grounds the 
infinity of space as actually existent, If he be an 
adherent of another school he may hold that the 
proposition " space is infinite" is incapable of 
proof, and that it can never be maintained ; but 
he will not on that account deny that he can con- 
ceive infinite space. One may maintain that our 
assent to the former proposition is conditioned 
on our assent to the latter; that if the infinite 
be so unattainable _and even contradictory a con- 
ception as Sir William Hamilton has held, we 
would have no reason to believe the existence of 
any infinite object either possible or actual; but 
certainly no one will hold that the first of the 
two propositions is so based upon the second as 
to necessarily stand or fall with it. The fact 
that I can imagine 

"... the Cannibals, that each other eat, 
The Anthropophagi, and men whose heads 
Do grow beneath their shoulders," 

does not prove such objects to have real exist- 
ence. If men were only able to represent in the 
imagination what has its actual prototype in 
nature, how would we account for 



THE CONCEIVABLE AND THE EXISTENT. 79 
"... the pert Fairies and the dapper Elves," 

and the cloud of unreal creatures with which 
the teeming poetic imagination of mankind has 
peopled the world from earliest ages ? Where 
would be the dragon, the basilisk, the roc? 
Where the valley of diamonds and the palace of 
Aladdin? The fact that I can conceive such does 
not prove that in the whole realm of nature such 
objects may be found. And, similarly, the fact 
that no objects of a certain kind have an actual 
existence is no proof that the various qualities 
representing them may not be found grouped in 
mind, as a result of the constructive imagination. 
I may examine in detail all actual horses and all 
actual men, and conclude that the qualities be- 
longing to each animal are distinct and sepa- 
rate ; but I cannot assure myself that some one 
will not form in his mind the notion of an ani- 
mal which combines the two sets of qualities, — a 
centaur. All our ideas are not derived in their 
ultimate shapes from direct intuition of objects, 
but are a result of mental processes, which recast 
and variously combine, the material furnished. 
Though we may not be able to add a new 
element to those simple factors of our experi- 
ence primarily given, yet it cannot be denied that 



80 THE CONCEPTION OF THE INFINITE. 

we may so transpose or combine, acid or elimi- 
nate, some of those elements, as to obtain a prod- 
uct consisting of elements which are not found 
thus combined in any single object as it is pre- 
sented to us in nature. When it is asked, there- 
fore, whether any given conception be conceiv- 
able, it is not necessary to an affirmative answer 
that any existent object be proved to correspond 
to the conception. It is enough if the qualities 
connotated by the name be shown to be such as 
may be mentally put together, and are not mutu- 
ally repugnant or contradictory. We find thus 
that the answer to the first question put forward 
at the beginning of the chapter does not depend 
on the answer to the second, and may be true 
though that be false. When it is asked whether 
the answer to the second question ,niay not be 
independent of the answer to the first, the an- 
swer is not so clearly affirmative; and although 
this point is not directly connected with the sub- 
ject under discussion, I may say, a projws of the 
antagonism which Hamilton exhibits between 
Faith and Knowledge, that, were the things of 
which I have just spoken so defined that I could 
in no conceivable manner put together in mind 
the contradictory qualities attributed to them, I 



THE CONCEIVABLE AND THE EXISTENT. 81 

should be doing rashly to assume their ex- 
istence, however veracious the witness upon 
whose word their acceptance might depend. It 
may be a question whether any force can be 
brought to bear upon a man sufficiently strong 
to lead him to believe in the actual existence of 
an inconceivable object; that is, of that which 
cannot become an object of thought, nor, conse- 
quently, of belief. However, the point in which 
we are now chiefly concerned is this : that there 
are a vast number of things that we have no 
reason to believe in as actually existent, but 
which we nevertheless conceive with perfect 
clearness. 

Again. When we put forward the proposi- 
tion that an infinite object is conceivable; 
unless we examine carefully this term used, — . 
conceivable, — and find out whether it has one 
meaning or two, we may involve ourselves in 
serious error. If the word has two meanings, 
or is in common use made to stand for two 
distinct mental operations ; when we prove that 
an infinite object is inconceivable in one ac- 
ceptation of the term, we do not necessarily 
prove that it is inconceivable in the other 
sense. If we suppose that in its first meaning 



82 THE CONCEPTION OF THE INFINITE. 

the word conceivable is synonymous with im- 
aginable, and that, when we eliminate one or 
more of the qualities which must be present 
to make an object imaginable, we may still in 
some way mentally grasp the remaining quali- 
ties, this second operation we may call con- 
ceiving in the second and narrower sense of the 
term. The proof that there is such a mental 
operation I will reserve for the next chapter. 

Let it be marked, therefore, that we have to 
consider three distinct propositions, which de- 
mand as many distinct kinds of proof, when 
we assert, in the first place, that an infinite ob- 
ject is conceivable, and in the second, that an 
infinite object exists. 

If I assert, then, that I can conceive space 
to be infinite, the proof should consist in an 
analysis of the qualities represented by the 
word, and an examination of their mutual con- 
sistency or repugnancy. If one element com- 
prehended in the conception that I have in 
mind be the notion of a certain quantity, then 
I must in some way mentally grasp that quan- 
tity and connect it with the other elements in 
the conception, as I do in the representation 
in imagination of finite objects. If there be in 



THE CONCEIVABLE AND THE EXISTENT. 83 

the conception no quantitative element, all that 
is necessary is the ability to grasp in thought 
certain qualitative elements, and this would fall 
under the second and narrower sense of the 
term conception. In either case the proposi- 
tion depends on no fact of actual being, and 
may be established without any reference to 
such. 

But the proposition " space is infinite" de- 
mands in the way of proof what the previous 
propositions can dispense with. Admitting my 
ability to conceive infinite space, I may yet 
doubt its existence,* which doubt I may pro- 
ceed to remove by proof. I may assume on 
a priori grounds the existence of infinite space, 
as a postulate of Eeason ; or I may deny the 
possibility of establishing the proposition a 
priori and be forced to fall back upon obser- 
vation, which proceeds by adding space to 
space. In the latter case, although I may ad- 
mit that the conception of the infinite is qual- 
itative, and means in the case in point only 
that a progression along a given line may be 



* I purposely avoid all consideration of metaphysical argu- 
ments drawn from theories as to the nature of space. 



84 THE CONCEPTION OF THE INFINITE. 

endless, yet the only apparent possibility of es- 
tablishing this fact with reference to any given 
line lies in a continued addition of parts. That 
is, though the conception itself may be quali- 
tative, and contain no reference to a whole and 
parts, the only way in which it seems possible 
to prove any actual object infinite is by em- 
ploying the ideas of quantity and totality in 
adding part to part, which ideas are in fact 
contradictory to ideas already contained in the 
conception. 

Now, it is evident that observation, adding 
part to part, could never attain the infinite, 
for the infinite time in which it is said to be 
possible to accomplish this is no quantity of 
time at all, and the phrase merely expresses in 
a disagreeably tautological and roundabout way 
that it can never be. But in spite of this, 
since our experience in going from part to 
part seems to be analogous to what would be 
our experience of the object if it were, indeed, 
infinite; and since there seems to be no other 
way of proving from experience its infinity, we 
are apt to imagine that by this process we are 
somehow proving the infinity of an object, or 
are at least in the way to prove it. Evidently 



THE CONCEIVABLE AND THE EXISTENT. 85 

this feeling was in the mind of Kant when he 
put forward the first of his antinomies ; and it 
is also evident that it exerted its influence on 
the mind of Hamilton, leading him to " rise 
from sphere to sphere in the region of empty 
space" in the fruitless endeavor to exhaust in- 
finite space. The same feeling it was that 
caused Clifford to affirm that, when we have 
counted one, two, three, four miles up into 
space, although we must admit that the attempt 
can never give us infinite space, we may yet 
regard our task as " distinctly begun." The 
attempt must fail, he thought, but this is the 
only way even to make the attempt. 

Eow, if Sir "William Hamilton had set before 
himself the task of proving that we can never 
know space to be infinite, his argument would 
have been to the point. If he denied the 
possibility of knowing space to be infinite a 
priori, no method of proving it such was left 
except the method of observation. He could 
not, of course, reasonably hold that the em- 
pirical method is a true method of proving 
anything to be infinite, but he might prove 
with the arguments that he has used for an- 
other purpose that the only even apparent 
8 



86 THE CONCEPTION OF THE INFINITE. 

method of proving space infinite is merely ap- 
parent, and that consequently we can never 
know space to be infinite. 

Or if Sir William had set before himself the 
task of proving that we cannot imagine or call 
up before the mind an exact and complete rep- 
resentation of an infinite line, his arguments 
would not have been aside from the point. 
The imagination can picture but a small ex- 
tent of line at any one time ; this portion must 
consequently be limited ; each portion succes- 
sively added in imagination must also have its 
limits; and we have no escape from the very 
difficulty which we have before met in the at- 
tempt to know an object as infinite, — the im- 
possibility of getting rid of the limits altogether. 
Whether we conceive our conception of an in- 
finite line to be quantitative or qualitative, 
there is an equal impossibility of getting rid 
of the quantitative in trying empirically to 
prove a line infinite and in imagining an infi- 
nite line. 

Sir William accepted this argument as a proof 
of the inconceivability of the infinite. Now, an 
argument which will prove any infinite object 
unknowable as an actually existing thing will 



THE CONCEIVABLE AND THE EXISTENT. 87 

prove an infinite object unimaginable only on 
the supposition that the element which has 
caused the impossibility in the former case be 
also present in the latter. An argument which 
would prove that I cannot know a line to be in- 
finite, because I cannot in the attempt to prove 
it such get rid of the element of quantity, would 
prove that I cannot imagine an infinite line, pro- 
vided that the attempt to imagine such a line 
must be similar in kind to the former process, 
and must meet the same difficulty with the 
quantitative element. And, similarly, an argu- 
ment which would prove an infinite line un- 
knowable and unimaginable on account of the 
difficulty occasioned by this quantitative element, 
would prove an infinite line inconceivable (in the 
narrower sense of the term) only if the element 
is necessarily present in conceiving (in the nar- 
rower sense) an infinite line that has caused the 
trouble in the former cases. 

But if we can prove that, when the notion of 
quantity is eliminated from a certain complex of 
psychic elements, the remaining elements can 
be in some way grasped in mind, and are thus 
grasped in certain actual and not infrequent 
mental operations, in this case the proofs, which 



88 THE CONCEPTION OF THE INFINITE, 

might be suitable when applied to the former 
propositions, can have no force whatever. 

"When we come in fact to make a more com- 
plete analysis of the elements comprehended in 
our conception of the infinite, we will see that 
the arguments which have been so often ad- 
vanced to prove the infinite inconceivable, and 
regarded as so unanswerable, are really not to 
the point at all, but may be classed under the 
old logical fallacy of ignoratio elenchi Sir Wil- 
liam Hamilton proves inconceivable something 
which is not at all supposed by the phrase " an 
infinite line," and which, in fact, contains an 
element flatly contradictory to one of those in- 
dicated by the phrase. 

In view of the foregoing, the exhibition of the 
danger in which we stand of confounding three 
distinct propositions and their appropriate proofs, 
the two points which I would insist upon are 
these : (1) That the conceivability of the infinite, 
in the narrower sense of the term, is quite dis- 
tinct from a knowledge of its existence, or from 
its imaginability, and it is with the possibility of 
the first alone that we have to do ; and (2) that, 
whatever be the metaphysical tenets embraced 
by one as to the nature of Space and Time, or 



THE CONCEIVABLE AND THE EXISTENT. 89 

of those universal laws of Being known by one 
school as Rational Intuitions and by an opposing 
school as Highest Generalizations from Expe- 
rience, — that these tenets, however they may 
lead to the acceptance or rejection of the propo- 
sition which affirms the existence of an infi- 
nite, can yet not aifect the proposition which 
affirms its conceivability. 



8* 



CHAPTEE VI. 

THE CONCEIVABILITY OF THE INFINITE. 

When we analyze the mental state in which 
we have reference to an infinite — let us take, 
for example, an infinite line — we find the fol- 
lowing elements : in the first place, there are 
present the usual qualities of a line; for the 
fact of our conceiving it as without limits need 
not alter any of its usual qualities, any more 
than the fact of our being unable to see the 
ends of a telegraph wire need force us to deny 
that it is a wire of a certain diameter, mate- 
rial, or color; and, in the second place, there 
is present the notion that, however far we may 
go in thought, we will find a continuation of 
the line. In other words, there is the notion of 
unlimited possibility of quantity, — a notion which, 
be it marked, is strictly qualitative. That 
there is no quantitative element present has 
been insisted upon in previous chapters. But 
quantity in general, not this or that quantity, 

is as much a qualitative notion as color or form; 
90 



THE CON CE1V ABILITY OF THE INFINITE. 91 

and in defining the second element present in 
our conception of an infinite line, I have used 
the word advisedly to bring out what is a dis- 
tinctive characteristic of the conception. The 
word infinite does not denote a quantity, but 
it has reference to quantity, and it cannot, in 
accordance with its derivation and true signifi- 
cation, be rightly applied to what is incapable 
of being quantitatively considered. My objec- 
tion to the usage of the word infinite, by some 
who recognize that the conception for which 
it stands is qualitative, is that they overlook 
the distinctive characteristic of this conception, 
which marks it out from other qualitative con- 
ceptions, — that is, its necessary reference to 
quantity, though not itself quantitative. If, by 
that process of abstraction which takes place 
when I compare objects similar in some of 
their qualities, I fix my attention upon the 
other qualities of any finite line, disregarding 
its length, and leaving out of view for the time 
being its limits, my conception is qualitative ; 
and yet it is not the conception of an infinite 
line. In this case, so far from affirming infi- 
nite length, I do not think of length at all. 
But, in the case of an infinite line, I add to 






92 THE CONCEPTION OF THE INFINITE. 

the former complex of qualities a new quality, 
possibility of quantity in general, not this or 
that quantity. When I try to bring before my 
mind the notion of an infinite line, what I am 
distinctly conscious of is this : I represent in 
imagination a line of indefinite length, and 
then run mentally along the line representing 
additional line-portions, — a proceeding which 
would of itself of course give me only the 
finite ; but what makes my conception distinc- 
tively of the infinite is that, in this progression 
or notion of continued increase, I fix my at- 
tention upon the progression itself, and elimi- 
nate by abstraction the limits to which such a 
progression is subject. I do not, be it marked, 
merely fix my attention upon the other quali- 
ties of a given line, abstracting from the notion 
of limits ; but I have in mind a progression, a 
possibility of ever-increasing quantity, and I 
abstract from the limits of this progression. 
The two conceptions are distinctly different, 
although both are qualitative, and they should 
not be confounded with one another. 

The question, therefore, whether I can con- 
ceive an infinite line is identical with the ques- 
tion whether I can mentally grasp the various 



THE CONCEIVABILITF OF THE INFINITE. 93 

qualities of a line, and the notion of a contin- 
ual increase of such a line, without including 
the notion of limits ; and it will be seen that 
this question is simply one of the phases of 
the broader question which is concerned with 
the possibility of the concept or general notion. 
A certain complex of qualities being necessary 
to the existence of a given object in nature, or 
to its subjective existence as represented in the 
imagination, is there any mental operation by 
which we may grasp some of these qualities, 
to the exclusion of others, and convey to our 
own and other minds by the use of the word 
which stands for this new complex a distinct 
meaning? 

There have been held with reference to this 
problem, as is well known, three opinions : the 
doctrine of the Eealists, that general ideas 
have corresponding to them a counterpart Re- 
ality, — a doctrine which may be passed over as 
now . abandoned, though its eiFects make them- 
selves felt in many directions; the doctrine of 
the Conceptualists, that, although general ideas 
cannot exist in Nature, nor be represented in 
the Imagination, yet they have a true mental 
existence, and are the result of a distinct men- 



94 THE CONCEPTION OF THE INFINITE. 

tal operation ; and the doctrine of the Nomi- 
nalists, that the only generality that has a 
separate existence, subjective or objective, is 
the Name, which may be applied indifferently 
to many similar objects. 

The Conceptualist may hold that it is possi- 
ble, unless the words include repugnant ele- 
ments, to conceive an infinite line, — that is, to 
grasp in mind a certain complex of psychic 
elements which are yet incapable of being 
pictured in the imagination as an infinite line. 
To think, in the sense of to form such a con- 
cept, is to him something other than to im- 
agine. What cannot be imagined may yet be 
thought. The word man, which we define as 
comprehending the elements of rationality and 
animality, he claims, does not in the least in- 
clude all those other qualities which must be- 
combined with these two before we can picture 
in the imagination or know as existing any 
given man. If we select the two qualities in 
which all the objects of a class resemble each 
other, and give to these two a special name, 
have we not brought them into consciousness 
in some way in which we have not the other 
qualities ? 



THE CONCEIVABILITY OF THE INFINITE. 95 

And when we turn to the Nominalist, it 
would not be hard to show that, although his 
doctrines, if taken in strictness, would deny 
the possibility of the mental operation by which 
we arrive at the concept, and consequently of 
the operation by which we may grasp in thought 
the various elements implied in the phrase " an 
infinite line," yet one may find in his teachings 
by implication ample justification for assuming 
its possible and actual existence. I will take 
some extracts from four well-known [Nominal- 
ists to show how palpable is the fact stated, 
and I will first quote from Berkeley, Locke's 
great opponent on the subject of the Abstract 
Idea : 

" Whether others have this wonderful fac- 
ulty of abstracting their ideas, they best can 
tell ; for myself, I find, indeed, I have indeed 
a faculty of imagining or representing to my- 
self the ideas of those particular things I have 
perceived, and of variously compounding and 
dividing them. I can imagine a man with two 
heads, or the upper parts of a man joined to 
the body of a horse. I can consider the hand, 
the eye, the nose, each by itself abstracted or 
separated from the rest of the body. But, 



96 THE CONCEPTION OF THE INFINITE. 

then, whatever hand or eye I imagine, it must 
have some particular shape and color. Like- 
wise the idea of man that I frame to myself 
must be either of a white, or a black, or a 
tawny, a straight, or a crooked, a tall, or a 
low, or a middle-sized man. I cannot by any 
effort of thought conceive the abstract idea 
above described. And it is equally impossible 
for me to form the abstract idea of motion 
distinct from the body moving, and which is 
neither swift nor slow, curvilinear nor recti- 
linear; and the like may be said of all other 
abstract general ideas whatsoever. To be plain, 
I own myself able to abstract in one sense, as 
when I consider some particular parts or quali- 
ties separated from others, with which, though 
they are united in some object, yet it is pos- 
sible they may really exist without them. But 
I deny that I can abstract from one another, 
or conceive separately, those qualities which it 
is impossible should exist so separated, or that 
I can frame a general notion by abstracting 
from particulars in the manner aforesaid, which 
last are the two proper acceptations of abstrac- 
tion. And there is ground to think most men 
will acknowledge themselves to be in my case. 



THE CONCEIV 'ABILITY OF THE INFINITE. 97 

The generality of men, which are simple and 
illiterate, never pretend to abstract notions. It 
is said they are difficult, and not to be attained 
without pains and study; we may, therefore, 
reasonably conclude that if such there be, they 
are confined only to the learned."* 

So much for Berkeley's position with respect 
to the abstract notion. But mark the conces- 
sions which he is forced to make in a later 
section : 

"But here it will be demanded, how we can 
know any proposition to be true of all partic- 
ular triangles except we have first seen it 
demonstrated of the abstract idea of a triangle 
which equally agrees to all ? For, because a 
property may be demonstrated to agree to 
some one particular triangle, it will not thence 
follow that it equally belongs to any other 
triangle, which in all respects is not the same 
with it. For example, having demonstrated 
that the three angles of an isosceles rectangular 
triangle are equal to two right ones, I cannot, 
therefore, conclude this affection agrees to all 

* "Principles of Human Knowledge." Introduction, Sect. 
10. Works, ed. Fraser, vol. i. pp. 141, 142. 



98 THE CONCEPTION OF THE INFINITE. 

other triangles which have neither a right 
angle nor two equal sides. It seems, there- 
fore, that to he certain this proposition is uni- 
versally true, we must either make a particu- 
lar demonstration for every particular triangle, 
which is impossible, or once for all demonstrate 
it of the abstract idea of a triangle, in which 
all the particulars do indifferently partake, and 
by which they are all equally represented. To 
which I answer, that though the idea I have 
in view whilst I make the demonstration be, 
for instance, that of an isosceles rectangular 
triangle whose sides are of a determinate length, 
I may nevertheless be certain it extends to all 
other rectilinear triangles, of what sort or 



bigness soever. And that because . neither the 
right angle, nor the equality, nor determinate 
length of the sides are at all concerned in the 
demonstration. It is true the diagram I have 
in view includes all these particulars, but then 
there is not the least mention made of them 
in the proof of the proposition. It is not said 
the three angles are equal to two right ones, 
because one of them is a right angle, or be- 
cause the sides comprehending it are of the 
same length. Which sufficiently shows that the 



THE CONCEIV 'ABILITY OF THE INFINITE. 99 

right angle might have been oblique and the 
sides unequal, and for all that the demonstra- 
tion have held good. And for this reason it is 
that I conclude that to be true of any obliquan- 
gular or scalenon which I have demonstrated 
of a particular right-angled equicrural triangle, 
and not because I demonstrated the proposi- 
tion of the abstract idea of a triangle. And 
here it must be acknowledged that a man may 
consider a figure merely as triangular without 
attending to the particular qualities of the 
angles or relations of the sides. So far he 
may abstract, but this will never prove that 
he can frame an abstract, general, inconsistent 
idea of a triangle. In like manner we may 
consider Peter so far forth as man, or so far 
forth as animal, without framing the fore- 
mentioned abstract idea either of man or of 
animal, inasmuch as all that is perceived is 
not considered."* 

In the former of the two extracts Berkeley 
has declared himself able to abstract only so 
far that he can represent to himself in im- 
agination what can exist separately in nature. 

-"Principles." Introduction, Section 16. 



100 THE CONCEPTION OF THE INFINITE. 

He denies that he can conceive separately those 
qualities which it is impossible should exist 
separately. But when he supposes an objector 
to ask how it is possible for something proved 
to be true of a particular triangle, to be known 
to be true of all triangles, he answers that it 
is seen that neither the right angle, nor the 
equality, nor the determinate length of the 
sides are at all concerned in the demonstra- 
tion. In other words, he admits that, so far 
as that demonstration goes, we have to do only 
with those elements in which all triangles agree. 
And if we can reason about certain elements to 
the exclusion of others ; if we can see that cer- 
tain objects are alike in certain elements and 
unlike in the others ; if we can give a name to 
objects simply to express the presence of these 
same elements, however the elements accom- 
panying them may vary, then surely the ele- 
ments of the concept have been before . the 
mind in some way in which the others have 
not, and have been grasped together. 

Berkeley frankly admits as much in the con- 
cluding sentences of the latter extract, sen- 
tences which were added twenty-four years 
after the first publication of the essay, when 



THE CON CEIV ABILITY OF THE INFINITE. 101 

mature reflection, we may suppose, had. brought 
him to see that on his previous principles, strictly 
held, all comparison of objects differing in any 
of their qualities would be impossible. If we 
can consider a figure merely as triangular, with- 
out attending to the particular qualities of the 
angles or relations of the sides, then we can 
in some sort divorce the elements included 
under the general word triangle from the ac- 
companying elements and consider them sepa- 
rately. In those last few sentences Berkeley 
admits all that a reasonable Conceptualist would 
care to prove, and the words " abstract idea," 
as there used, are equivalent to " object of the 
imagination," a something which is not implied 
in the formation of the abstract or general 
notion. 

Every one, Nominalist or Conceptualist, must 
acknowledge that we can compare objects and 
recognize them as like or unlike, — not merely 
like or unlike as wholes, but in this or that 
element; like in length, unlike in breadth; like 
in color, unlike in shape. Now, no one claims 
that we can call into clear consciousness the 
element of length alone, and picture it divorced 
of breadth and color; but when we recognize 

9* 



102 THE CONCEPTION OF THE INFINITE. 

two objects as like in length and unlike in 
breadth, the elements must in some way have 
been present in mind separately, so as to be 
i recognized as length and breadth. If one object 
that what is present in consciousness must ipso 
facto be perceived, and that we cannot perceive 
length as a factor by itself, nor recall in memory 
any perception of such a factor during the act 
of comparison, I answer that what is in con- 
sciousness is by no means necessarily in a clear 
analytic consciousness, and that we may by a 
process of deductive reasoning be sure that 
certain elements are present as factors in a 
given mental state, while we are yet quite un- 
able to call these elements into a clear analytic 
consciousness, separated from certain other 
elements bound to them by long association 
and habit. As an instance, I refer to vision. 
That distance is itself unperceivable by sight 
we must admit. That judgments of distance 
are a result of reasoning from an observed 
constant connection of certain visual with cer- 
tain other elements, may be satisfactorily es- 
tablished when the above proposition is ad- 
mitted. But to call into clear consciousness 
by itself the purely visual sensation, which 



THE CONCEIVABILITY OF THE INFINITE. 103 

forms the basis of the judgment, is altogether 
impossible. That it is a factor, and an im- 
portant factor, in the complex consciousness 
which we have at the time, we know, and yet 
its presence, as a single and distinct element, 
is capable of being only deductively known. 
Notice a further point which is worthy of re- 
mark. If we vary the purely visual element, 
allowing all the other elements to remain the 
same, — that is, if we change the color of the 
object, but do not change in any respect the 
form or size of the image on the retina, — a 
difference is at once remarked, and the change 
of color recognized. But the difference is not 
recognized as a difference between two purely 
visual sensations when the result of the actual 
comparison comes into clear consciousness, but 
as a difference in one of their elements between 
two complexes or wholes. That is to say, the 
two visual sensations cannot be brought into 
clear consciousness and compared with each 
other alone, but only as they" are connected 
with certain other elements in complexes or 
wholes; it is the presence of two or more such 
wholes, which we wish to compare, that pri- 
marily impels to the narrowing of the atten- 



104 THE CONCEPTION OF THE INFINITE. 

tion to the single similar or dissimilar elements. 
This point is specially worthy of remark, as 
there is something closely analogous to this in 
the formation of the concept in general, and 
this special case may help to throw light upon 
all cases in which that which cannot be im- 
agined is yet thought. 

"When I form the concept of length by com- 
paring two objects in length and affirming 
agreement, and then recognizing as a distinct 
element that in which they agree, I certainly 
do not compare the objects simply as wholes, 
but compare the lengths; and I must surely 
have had these elements in mind in some way 
in which I had not the other elements which 
go to make up the object. Whether I can call 
into clear consciousness the psychic elements 
present during the operation or not, it does 
not much matter. I evidently have specialized, 
selected some elements from among others, and 
compared length with length, as element with 
element. The name which we give to such 
resemblances is the name representing a gen- 
eral or abstract idea. Whether the possibility 
of thus comparing single elements may not be 
always conditioned by the presence of two or 



THE CONCEIVABILITY OF THE INFINITE. 105 

more objects or complexes in which the ele- 
ments are present I will consider later. 

Hume warmly applauds the position taken 
by Berkeley with reference to the abstract 
idea, calling it " one of the greatest and most 
valuable discoveries that have been made of 
late years in the republic of letters," and he 
undertakes to confirm it with proofs that he 
hopes will put it " beyond all doubt and con- 
troversy." For the same purpose for which 
I quoted the two extracts from Berkeley, I 
will quote the last part of the section which 
he devotes to the establishment of this posi- 
tion : 

" It is certain that the mind would never 
have dreamed of distinguishing a figure from 
the body figured, as being in reality neither 
distinguishable, nor different, nor separable, did 
it not observe that even in this simplicity there 
might be contained many different resemblances 
and relations. Thus, when a globe of white 
marble is presented, we receive only the im- 
pression of a white color disposed in a certain 
form, nor are we able to separate and distin- 
guish the color from the form. But observing 
afterwards a globe of black marble and a cube 



106 THE CONCEPTION OF THE INFINITE. 

of white, and comparing them with our former 
object, we find two separate resemblances in 
what formerly seemed, and really is, perfectly 
inseparable. After a little more practice of this 
kind we begin to distinguish the figure from 
the color by a distinction of reason, — that is, we 
consider the figure and color together, since 
they are, in effect, the same and undistinguish- 
able, but still view them in different aspects, ac- 
cording to the resemblances of which they are 
susceptible. When we would consider only the 
figure of the globe of white marble, we form in 
reality an idea both of the figure and color, 
but tacitly carry our eye to its resemblance 
with the globe of black marble; and in the 
same manner, when we would consider its 
color only, we turn our view to its resemblance 
with the cube of white marble. By this means 
we accompany our ideas with a kind of reflec- 
tion, of which custom renders us, in a great 
measure, insensible. A person who desires us 
to consider the figure of a globe of white mar- 
ble without thinking on its color, desires an 
impossibility; but his meaning is that we 
should consider the color and figure together, 
but still keep in our eye the resemblance to 



THE CON CEIV ABILITY OF THE INFINITE. 107 

the globe of black marble, or that to any other 
globe of whatever color or substance."* 

It is not hard to see that we cannot distin- 
guish in a body figured " many different re- 
semblances and relations" without bringing the 
resembling elements in some sense singly into 
thought; if the mental complex which we call 
an object were an indissoluble unit, we could 
affirm a general likeness or unlikeness between 
it and other objects, but we could not affirm 
that the resemblance lay in the figure or color. 
If, as Hume asserts, the figure and color " are, 
m effect, the same and undistinguishable," why 
do we find the one susceptible of the one class 
of resemblances and the other of another class ? 
If we take the words literally, should not the 
figure, viewed in one aspect, be susceptible of 
resemblances of figure, and viewed in another 
of color? And, similarly, if the color is one 
with the figure, — the same and undistinguisha- 
ble, — should not the color, viewed in one aspect, 
be susceptible of resemblances of color, and 
viewed in another of figure ? Hume's admis- 

*" Treatise of Human Nature," bk. i. Sect. 7. "Works, 
Boston, 1854, vol. i. p. 42. 



108 THE CONCEPTION OF THE INFINITE. 

sion that the two elements are known as giving 
different resemblances, in itself refutes his pre- 
vious assertion that they are undistinguishable. 
If color be recognized as like color, and figure 
like figure, the two qualities are distinguished 
as different, and are in reality separately 
grasped. 

I will now take a passage from Mr. J. S. 
Mill's "Examination of Sir William Hamil- 
ton's Philosophy" : 

" The formation, therefore, of a Concept 
does not consist in separating the attributes 
which are said to compose it from all other 
attributes of the same object, and enabling us 
to conceive those attributes, disjoined from 
any others. We neither conceive them, nor 
think them, nor cognize them in any way as a 
thing apart, but solely as forming, in combina- 
tion with numerous other attributes, the idea 
of an individual object. But, though thinking 
them only as part of a larger agglomeration, 
we have the power of fixing our attention on 
them to the neglect of the other attributes 
with which we think them combined. While 
the concentration of attention actually lasts, 
if it is sufficiently intense, we may be tempo- 



THE C0NCE1V ABILITY OF THE INFINITE. 109 

rarily unconscious of any of the other attri- 
butes, and may really, for a brief interval, 
have nothing present to our mind but the at- 
tributes constituent of the concept. In gen- 
eral, however, the attention is not so com- 
pletely exclusive as this ; it leaves room in 
consciousness for other elements of the con- 
crete idea; though of these the consciousness 
is faint in proportion to the energy of the con- 
centrative effort, and the moment the attention 
relaxes, if the same concrete idea continues to 
be contemplated, its other constituents come 
out into consciousness. General concepts, there- 
fore, we have, properly speaking, none; we 
have only complex ideas of objects in the con- 
crete ; but we are able to attend exclusively to 
certain parts of the concrete idea ; and by that 
exclusive attention we enable those parts to de- 
termine exclusively the course of our thoughts 
as subsequently called up by association, and 
are in a condition to carry on a train of medi- 
tation or reasoning relating to those parts only, 
exactly as if we were able to conceive them 
separately from the rest."* 

*" Examination of Sir William Hamilton's Philosophy," 

vol; ii. p. 64. Boston, 1868. 

10 



HO THE CONCEPTION OF THE INFINITE. 

This passage is so clearly in harmony with, 
the views of the Conceptualist, as I have por- 
trayed them, that it seems scarcely necessary 
to comment upon it. But I cannot resist the 
temptation to delay for a moment over an in- 
consistency into which Mill was forced by his 
attempt to recognize, though a ^Nominalist, a 
truth which the Nominalist, pure and simple, 
cannot recognize. The formation of a Concept, 
lie insists, does not consist in " separating the 
attributes said to compose it from all other 
attributes of the same object, and enabling us 
to conceive those attributes, disjoined from any 
others." This position he emphasizes by the 
further affirmation that " we neither conceive 
them, nor think them, nor cognize them in any 
way as a thing apart, but solely as forming, in 
combination with numerous other attributes, 
the idea of an individual object." These sen- 
tences are certainly unequivocal : they con- 
tain an emphatic assertion of the nominalistic 
doctrine. 

But, side by side with such statements, we 
find it asserted that we may fix the attention 
upon the attributes constituent of the concept, 
to the neglect of the other attributes of the 



THE CON CEIV ABILITY OF THE INFINITE, m 

object, and that while the concentration of at- 
tention actually lasts, if it is sufficient!} 7 intense, 
" we may be temporarily unconscious of any of 
the other attributes, and may really, for a brief 
interval, have nothing present to our mind but 
the attributes constituent of the concept." 
Surely, if the only elements before the mind 
are those constituent of the concept; if we 
may be conscious of these, even for a brief 
interval, and conscious of these alone; surely 
in such a case we conceive, or think, or in some 
way cognize the attributes forming the concept 
as separate and apart, and not for the time 
being, in combination with numerous other 
attributes. Mr. Mill goes even further in the 
above admission than most of us would care to 
follow him. In speaking as he does of the 
process, and not distinguishing between the 
imagining of an object and the knowing of one 
or more of its isolated qualities, he clearly in- 
timates, although he does not distinctly say, 
that the elements before the mind during the 
formation or use of a concept are in conscious- 
ness in the same way that the whole complex or 
object may be in consciousness. But, to recur 
to the before-mentioned analogy of the purely 



112 THE CONCEPTION OF THE INFINITE. 

visual element in vision, we know that, although 
we may so concentrate the attention as to dis- 
tinguish the blue color of one object from the 
red color of another, and so must have com- 
pared in some rapid manner these purely visual 
sensations, yet when we try to call into clear 
consciousness the mere sensation of color, we 
cannot do it without imagining the color as 
on a surface, or combined with psychic ele- 
ments not purely visual. That is to say, the 
single and separate sensations cannot be called 
into a clear consciousness, and their presence 
when we use the concept, or have occasion to 
compare them singly with each other, is some- 
thing quite distinct and different from the pres- 
ence in consciousness of the complex which is 
knowable as an object. And such would seem 
also to be the case wherever we call before 
the mind the single psychic elements which can 
yet not be represented alone in the imagination. 
The element must have been grasped separately, 
but it can be brought into a clear consciousness 
only in combination. 

If now we recognize in each of two objects 
presented to us a certain quality or complex 
of qualities upon which we can fix the atten- 



THE CONCEIVABILITY OF THE INFINITE. H3 

tion, and if we discover that, so far as these 
qualities go, there is an undistinguishable sim- 
ilarity in the objects, the differences arising al- 
together from other qualities, why may we not 
call the complex of qualities in point a general 
notion or general idea ? Of course, whether we 
should call the qualities in the two instances 
the same, even if they were undistinguishably 
similar, would depend on our use of the word 
same,* and our ideas of what constitutes same- 
ness or identity; but I can see no objection to 
using the words " general notion" to indicate 
the fact that a certain complex of qualities is 
to be found in many different combinations 
with other qualities. Should it still be insisted 
that, since we cannot bring separately into 
clear consciousness these elements of objects 
known, we have no reason to assume that 
we actually conceive them or think them sep- 
arately, I will not quarrel over the use of 
a word, but will simply state that I find the 
word " conceive" a useful one to express that 
concentration of the attention upon certain 



* I have pointed out before the fact that the word 
same" is commonly used in four quite distinct senses. 
10* 



114 THE CONCEPTION OF THE INFINITE. 

qualities of an object which takes place when 
objects are compared, and which eliminates 
from consciousness, or at least subordinates 
all other qualities of the objects; and I will so 
use the word, applying it to an operation the 
existence of which Mr. Mill has in so many 
words admitted. 

The last author whom I will quote is Mr. 
Bain. I will take some passages from the 
chapter on abstraction in his compendium of 
psychology and ethics, where he supports the 
Nominalistic doctrine : 

" We are able to attend to the points of 
agreement of resembling things, and to neg- 
lect the points of difference, as when we think 
of the light of luminous bodies, or the round- 
ness of round bodies. This power is named 
Abstraction. 

"It is a fact that we can direct our attention 
or our thoughts to the points of agreement 
of bodies that agree. "We can think of the 
light of the heavenly bodies, and make asser- 
tions, and draw inferences respecting it. So 
we can think of the roundness of spherical 
bodies, and discard the consideration of their 
color and size. In such an object as the full 



THE CONCEIV 'ABILITY OF THE INFINITE. H5 

moon we can concentrate our regards upon its 
luminous character, wherein it agrees with one 
class of objects, or upon its figure, wherein it 
agrees with another class of objects. We can 
think of the taste of a strawberry, either as 
agreeing with other tastes or as agreeing with 
pleasures generally. . . ." 

" Every Concrete thing falls into as many 
classes as it has attributes ; to refer it to one 
of these classes, and to think of the corre- 
sponding attribute, are one mental operation. 

" When a concrete thing before the view re- 
calls others agreeing in a certain point, our at- 
tention is awake upon that point; when the 
moon recalls other luminous bodies, we are 
thinking of its light; when it recalls other 
round bodies, we are thinking of its roundness. 
The two operations are not different but iden- 
tical . 

" On this supposition, to abstract, or to think 
of a property in the abstract, is to classify 
under some one head. To abstract the prop- 
erty of transparency from water is to recall, 
at the instance of water, window-glass, crystal, 
air, &c. ; to abstract its liquidity is to recall 
milk, vinegar, melted butter, mercury, &c. ; to 



116 THE CONCEPTION OF THE INFINITE. 

abstract its weight is to bring it into compari- 
son with other kinds of gravitating matter. 

" Hence abstraction does not properly con- 
sist in the mental separation of one property 
of a thing from the other properties, as in 
thinking of the roundness of the moon apart 
from its luminosity and apparent magnitude. 
Such a separation is impracticable ; no one can 
think of a circle without color and a definite 
size. All the purposes of the abstract idea 
are served by conceiving a concrete thing in 
company with others resembling it in the at- 
tribute in question, and by affirming nothing 
of the one concrete but what is true of all 
those others." 

..." In abstract reasoning, therefore, we 
are not so much engaged with any single 
thing as with a class of things. When we 
are discussing government, we commonly have 
in view a number of governments alternately 
thought of; if we notice in any one gov- 
ernment a certain feature, we run over the 
rest in our mind, to see if the same feature is 
present in all. There is no such thing as an 
idea of government in the abstract; there is 
only possible a comparison of governments in 



THE CONCEIVABILITY OF THE INFINITE. H7 

the concrete; the abstraction is the likeness or 
community of the individuals."* 

It will be noticed that throughout this ex- 
tract Mr. Bain does not distinguish between 
the elements of an act which come out into a 
clear consciousness and the elements which do 
not so come out, but are necessary to the possi- 
bility of the operation. When Mr. Bain says, 
for instance, that there is no such thing as the 
idea of government in the abstract, but that 
we can compare governments in the concrete, 
and recognize the likeness of the individuals, 
it is perfectly true that all that we are clearly 
conscious of is several individual objects and 
a similarity between them ; but when we come 
to analyze this recognition of a similarity, it 
will be seen that the elements which are known 
as similar are quite incapable, by themselves, 
of forming a concrete object, and yet they are 
distinguished by the mind from the dissimi- 
lar elements; they must, therefore, have been 
in some sort grasped separately, though they 
cannot separately be brought into a clear con- 
sciousness. Whether, during this rapid act of 

* " Mental and Moral Science," London, 1868, pp. 176-78. 



118 THE CONCEPTION OF THE INFINITE. 

concentration and comparison, the other ele- 
ments which go to form the object actually 
disappear from consciousness, or are only dimly 
perceived, as Mr. Mill suggests that they are in 
most cases, does not affect the peculiar char- 
acter of the act. When I compare in height 
two trees which I see side by side in the dis- 
tant landscape before me, I am perhaps con- 
scious of several objects in their immediate 
vicinity in a dim and indefinite way, but the 
two objects compared are present in con- 
sciousness in a manner very different, and are 
grasped, so to speak, separately. And when I 
fix my attention upon the height of the two 
trees, finding them similar or dissimilar in this 
one element, we have every reason to suppose 
that something very analogous takes place, 
and that this one element is present in con- 
sciousness in some way quite different from 
the others, and is grasped separately for the 
time being. Were it not so, we could not 
say the trees are alike in height, but different 
in contour or color of foliage. We are justi- 
fied in assuming that when we recognize two 
trees as like in height, but not in color, we 
have compared height with height and color 



THE CONCEIVABILITY OF THE INFINITE. H9 

with color, and not merely compared the one 
object as an undistinguishable complex of qual- 
ities with another object as another undis- 
tinguishable complex. 

As I have before said, the name which we 
choose to apply to this operation is of little 
consequence ; the point chiefly to be borne in 
mind is that we have here an operation differ- 
ing from ordinary imagination, in that it takes 
cognizance of certain psychic elements which 
can yet not be called into clear consciousness 
by themselves as a mental picture. Whether 
the two operations completely differ in their 
ultimate nature is another question. When 
the Conceptualist asserts that though he can- 
not imagine length apart from breadth or 
color, yet he can conceive or think it, he 
merely marks by a distinct name his recogni- 
tion of an operation different from imagina- 
tion, and which is implied in all comparison 
of objects. What may be the peculiar psychic 
elements present in the operation he does not 
necessarily know, nor express when he uses the 



* Kant seems to have despaired of the possibility of ever 
making this analysis: " Dieser Schematismus unseres Ver- 



120 THE CONCEPTION OF THE INFINITE. 

Arguing from the analogy of the purely 
visual element in vision, one might conclude 
that what is actually present in consciousness 
in comparing lengths, for instance, is the dis- 
tinctive element which is present in combina- 
tion with other elements (and, consequently, in 
a modified form) in all our experience of ex- 
tended objects, but which, in the act of com- 
paring two objects, may be brought into suf- 
ficient prominence to be considered, for the 
moment, alone, and alone compared with its 
kind. When we make the attempt to call it 
into clear consciousness, the element appears as 
modified by, and in combination with, others ; 
but it is not improbable that, in the act of 
comparison, it obtains in its pure state suffi- 
cient recognition to make possible a comparison 
with a similar element also in its pure state. 
However, whether we can describe just what 

standes, in Ansehung der Erscheinungen und ihrer blossen 
Form, ist eine verborgene Kunst in den Tiefen der mensch- 
lichen Seele, deren wahre Handgriffe wir der Natur 
schwerlich jemals abrathen und sie unverdeckt vor Augen 
legen werden" (" Kritik, von dem Schematismus der 
reinen Verstandesbegriffe"), but he did not doubt the 
operation. 



THE CON GFAV ABILITY OF THE INFINITE. 121 

is present during the act or not, we may be 
sure that a mental separation of two objects 
into their elements is necessary in order to 
a recognition of them as in some points simi- 
lar and in some dissimilar. 

Hume has asserted that if we knew but the 
one object, and had no other objects with 
which to compare it, we would never distin- 
guish between the several elements composing- 
it; and the same thought is made prominent 
by Mr. Bain when he states that to refer an 
object to a class of other objects, and to think 
of the corresponding attribute, are but one 
act. Mr. Bain's statement is, of course, some- 
what inaccurate, as it confounds two very dif- 
ferent parts of the one operation ; but the point 
upon which both of these writers insist, — 
namely, that a comparison of objects is neces- 
sary to an analysis of the objects into their 
resemblances, — what should be called their ul- 
timate elements, — is worthy of attention. In 
all probability, were it not for a comparison 
of objects, a constant experience of groups of 
psychic elements containing likenesses and un- 
likenesses, we should never analyze the groups 
and make prominent single elements, separated 



122 THE CONCEPTION OF THE INFINITE. 

from their accompaniments. And since the 
single elements do not themselves come out 
into a clear analytic consciousness, it is not easy 
to see how we could be sure that they had been 
separately grasped, if we could not infer their 
presence from the possibility of comparison and 
classification of objects. That this analysis im- 
plies the presence of two objects, is necessarily 
a classification, after the analytic habit has been 
formed, as Mr. Bain insists, is not so clear ; but 
that, as a preliminary to the act of concentra- 
tion by which we form the concept, we call 
into consciousness at least one concrete object, 
I think, cannot be doubted ; and this fact might 
easily mislead one into taking the JSTominalistic 
position that a recognition of particular objects 
expresses the whole process. Particulars must 
be present of course, if they are to be compared; 
but, when compared, they are not compared as 
wholes. 

In view of the foregoing, I would, therefore, 
regard the fact as beyond all doubt, that there 
are mental operations differing distinctly from 
imagination, in that certain elements, of which 
we have usually, as single elements, no ana- 
lytic consciousness, but which are merged with 



THE CON CEIV ABILITY OF THE INFINITE. 123 

others into an indivisible whole, are brought for 
the time being into such prominence as to be 
compared individually with similar elements, and 
recognized as like or unlike. It does not follow 
that we may have a clear consciousness of the 
steps in the rapid process in which this compari- 
son takes place, or clearly recognize the nature 
of the elements compared; but from the fact of 
the comparison, about which there can be no 
doubt, we may be very sure that the operation 
in question has taken place. 

Now, when we return to the particular con- 
ception which we have been considering, — that 
of an infinite line, — we find it merely a con- 
crete instance of this general truth, which all 
must either explicitly or implicitly admit. As 
I have said, the elements constituent of this 
conception are the usual qualitative attributes 
of a line and the notion of continued pro- 
gression, of unlimited possibility of quantity. 
These elements may be brought into mind, to 
the exclusion of the notion of limits, which 
are yet present in all imagined lines and in 
all intuitions of lines in nature, by employing 
the process usual in forming a concept. When 
I think of an infinite line, I first represent to 



124 THE CONCEPTION OF THE INFINITE. 

myself a line of some indefinite length, and 
I then run mentally along this line, adding 
new portions, — that is, I successively think 
several increasing lengths. I have now before 
my mind what Hume and Bain insist upon 
and make so prominent in forming the con- 
cept, several concrete objects similar in some 
of their qualities. Having mentally passed 
over several of these line portions, I then fix 
my attention, not upon that in which they 
differ, — the quantitative element, — but upon 
that in which they resemble, the usual quali- 
tative attributes of a line, and the notion of 
increase or progression, which is common to 
all. This is precisely what I do in forming 
the concept man or animal. The concrete ob- 
jects are compared, their differences eliminated 
by abstraction, and their likenesses grasped to- 
gether under a distinctive name. Or I may 
select one of the qualities in which objects 
agree, and consider it alone, as when I com- 
pare men of the same age and color, only as 
to their height, and pronounce them equal in 
height. If this be possible, if, in using the 
word man, I can distinguish between that in 
which men agree and that in which they clisa- 



THE CONCEIVABILITV OF THE INFINITE. 125 

gree, and if it be further possible for me to fix 
my attention upon one of the points in which 
they agree, to the exclusion of others, then it 
is possible to abstract from the particular quan- 
tities or amounts of several lines present in im- 
agination, and think only of a constant increase 
or progression. That both the one and the 
other are not only possible but actual opera- 
tions, is proved beyond possibility of doubt by 
our constant comparison of objects, our use of 
general language, our frequent use of the word 
infinite, to indicate what is clearly distinguished, 
readily defined, and conveys a distinct meaning 
to speaker and hearer. 

One point I will here remark upon before 
passing, and that is the distinction sometimes 
drawn between the abstract and the general 
notion, a point at which I have a few pages 
before briefly hinted. Admitting that Hume 
was right in saying that had we not had oc- 
casion to compare two objects we should never 
have analyzed either into its elements, the ques- 
tion naturally arises, whether, after we have 
formed this habit of analysis by comparison, we 
may not by mere effort of will fix the attention 
upon one element of a single object, without any 



126 THE CONCEPTION OF THE INFINITE. 

reference to its occurrence in another object? 
To take an example, can I not, in imagining a 
window or a door, fix my attention upon its 
length, without thinking of the length of any- 
thing else, or comparing the object with any 
other extended object? 

The question cannot be answered off-hand, 
as by saying that in recognizing my perception 
as a perception of length — in using the word 
length — I necessarily class the object with 
other long objects; for it is at least thinkable 
that I may have so associated the word with 
this peculiar element as to have it suggested 
by the presence of the element, and still may 
not be conscious of other combinations in 
which the element occurs. It is, I think, 
highly probable, however, that when we con- 
centrate attention upon one element of an 
object, there is a more or less dim and vague 
reference to other objects, and that there is a 
rapid comparison ; but this fact must be proved 
by an interrogation of consciousness during the 
act, and upon this point I will not insist. If 
it be allowed that the one element may be 
known without reference to its occurrence in 
two or more objects, we have what may justly 



THE CONCEIV 'ABILITY OF THE INFINITE. 127 

be called the abstract notion, as distinguished 
from the element recognized as present in sev- 
eral combinations, in which latter case we may 
call it the general notion. And even if we 
deny that the abstraction is possible except 
there be two or more objects present in mind, 
and a comparison of them, yet it must be ac- 
knowledged that the prominence of these ob- 
jects in consciousness varies greatly, and, ac- 
cordingly, we may have either the intension or 
the extension of the concept most prominently 
before the mind. If we are concerned, not so 
much with the combinations in which the ele- 
ment occurs as with the element itself, we 
may call our notion abstract; if we have prom- 
inently in mind the number of occurrences, we 
may call it general. In either case the dis- 
tinction between the abstract and the general 
notion is a legitimate one. 

This point is not one directly connected 
with the subject with which I am concerned, 
— the conceivability of the infinite, — but is one 
too interesting to be overlooked in any exami- 
nation into the nature of the concept ; and, 
indeed, it can scarcely be considered totally 
foreign to the subject in hand, as the operation 



128 THE CONCEPTION OF THE INFINITE. 

of forming a concept, and the act of conceiving 
an infinite, are not different in their nature, and 
may be viewed in the same aspects. And to 
the objection which may be made to my class- 
ing the notion of this or that particular in- 
finite line with the concept or general notion, 
as I have done throughout this chapter, the 
objector taking the ground that the individual 
or the intuition is something quite different 
and distinct from the concept, — to this objec- 
tion I answer that the notion of any particular 
infinite line is not a complete intuition, in that 
one of the elements of the intuition is elimi- 
nated by abstraction; and that when, in the for- 
mation of any concept, we fix the attention 
upon certain elements of an intuition to the 
exclusion of others, we have in mind, so to 
speak, a constituent part of an intuition : the 
fact that we recognize its similarity, or, if we 
so choose to use the word, its sameness with 
parts of other intuitions, does not alter the 
individual character of the elements which we 
actually have in mind. The operation of form- 
ing a concept and the operation of conceiving 
an infinite line are in nature identical. 

It seems impossible that any one, having 



THE CONCEIVABILITY OF THE INFINITE. 129 

reflected upon the fact of his constantly grasp- 
ing in concepts elements which can yet not be 
separately imagined, and having, after an anal- 
ysis of what is in his mind when he calls up 
the notion of an infinite, discerned the iden-' 
tity of this latter operation with the former, it 
seems impossible that such an one should hold 
an infinite line, or infinite time or space to be 
inconceivable. But, being loath to give up his 
former position, such a man will probably put 
forward in a new form an objection upon which 
I have already commented. We have heard 
him object, "If we do not know the infinite 
as a whole, do we not know only its parts, 
which are finite ?" And now we will hear 
him object that, " Even if it be true that we 
can grasp in thought the notion of progression, 
and the notion of a line in general, this will 
give us no knowledge of an infinite line, but 
will give us only the elements of an incomplete 
image, which cannot be called distinctly before 
consciousness, and therefore cannot be known 
as an object at all." If, however, one feel 
himself aggrieved because he cannot represent 
to himself, endowed with all the qualities neces- 
sary to an object of the imagination, that which 



130 THE CONCEPTION OF THE INFINITE. 

he has already defined as wanting some of 
those qualities, he will be unreasonable enough 
to think it ground for complaint that he can- 
not in thought make parallel lines meet, or 
imagine a triangle with four sides. The word 
infinite means devoid of limits, and it neces- 
sarily follows that an infinite line cannot be 
known as a quantity, consequently not as a 
whole. Every object which is seen or imag- 
ined has necessarily limits, definite or indefi- 
nite: an infinite line, as infinite, cannot become 
an object of the imagination. But from this it 
by no means follows that I cannot call a par- 
ticular line infinite, provided I have some proof 
of the fact other than its conceivability, and 
that I cannot know my conception to be in 
harmony with the reality. Suppose that, either 
from testimony or by means of some a priori 
chain of reasoning, I have good reason to be- 
lieve a given line endless, I can conceive the 
line as without end, and I may know my con- 
ception, although it does not represent the 
total content of my consciousness when at any 
moment I gaze upon this or that part of the 
line, to be a true and real conception, and in 
harmony with my experience as I progres- 



THE CONCEIVABILITY OF THE INFINITE. 131 

sively pass over the line ; and I may be cer- 
tain that, however long my experience may 
continue, it will yet not prove incompatible 
with the conception I have formed. In this 
sense, and in this sense alone, is any infinite 
object conceivable, and there is no other con- 
ceivable way in which we could conceive it. 
An infinite object which could be known as a 
whole is not even an object of thought, for 
the elements indicated by the words cannot be 
so put together as to express a meaning. But 
the conception of the infinite, as I have de- 
fined it, contains in it nothing either contra- 
dictory or beyond the grasp of the human 
mind, and is, indeed, a very common concep-' 
tion, as is evidenced by use of the word infi- 
nite in literature, ancient and modern, to say 
nothing of the constant occurrence of the word 
in the debates of those very philosophers who 
find the conception such a stone of stumbling. 
And that the conception is a real one, having 
a real consonance with experience, those of us 
who hold to the Christian doctrine of Immor- 
tality will not be slow to maintain. 



